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Chapter 6: Proportions and Similarity

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Proportions and
Similarity
• Lessons 6-1, 6-2, and 6-3 Identify similar
polygons, and use ratios and proportions to solve
problems.
• Lessons 6-4 and 6-5 Recognize and use
proportional parts, corresponding perimeters,
altitudes, angle bisectors, and medians of similar
triangles to solve problems.
• Lesson 6-6 Identify the characteristics of
fractals and nongeometric iteration.
Similar figures are used to represent various real-world
situations involving a scale factor for the corresponding
parts. For example, photography uses similar triangles
to calculate distances from the lens to the object and
to the image size. You will use similar triangles to solve
problems about photography in Lesson 6-5.
280 Chapter 6 Proportions and Similarity
David Weintraub/Stock Boston
Key Vocabulary
•
•
•
•
•
proportion (p. 283)
cross products (p. 283)
similar polygons (p. 289)
scale factor (p. 290)
midsegment (p. 308)
Prerequisite Skills To be successful in this chapter, you’ll need to master
these skills and be able to apply them in problem-solving situations. Review
these skills before beginning Chapter 6.
For Lesson 6-1, 6-3, and 6-4
Solve Rational Equations
Solve each equation. (For review, see pages 737 and 738.)
2
1. бЋЏбЋЏy ПЄ 4 П­ 6
3
5
xПЄ4
2. бЋЏбЋЏ П­ бЋЏбЋЏ
6
yП©2
yПЄ1
2y
4
4
3. бЋЏбЋЏ П­ бЋЏбЋЏ
12
3
32
4. бЋЏбЋЏ П­ бЋЏбЋЏ
For Lesson 6-2
y
Slopes of Lines
Find the slope of the line given the coordinates of two points on the line.
(For review, see Lesson 3-3.)
5. (3, 5) and (0, ПЄ1)
6. (ПЄ6, ПЄ3) and (2, ПЄ3)
7. (ПЄ3, 4) and (2, ПЄ2)
For Lesson 6-5
Show Lines Parallel
Given the following information, determine whether a К€ b.
State the postulate or theorem that justifies your answer.
a
(For review, see Lesson 3-5.)
1
8. Р„1 РҐ Р„8
5
2
6
b
3
7
4
8
c
9. Р„3 РҐ Р„6
10. Р„5 РҐ Р„3
For Lesson 6-6
Evaluate Expressions
Evaluate each expression for n П­ 1, 2, 3, and 4.
(For review, see page 736.)
12. n2 ПЄ 2
11. 2n
13. 3n ПЄ 2
Proportions and Similarity Make this Foldable to help you organize your notes.
Begin with one sheet of 11” by 17” paper.
Punch and Fold
Divide
Open the flap and draw
lines to divide the inside
into six equal parts.
Fold widthwise. Leave
space to punch holes so it
can be placed in your
binder.
Label
Label each part using the
lesson numbers.
6-1
6-2
6-3
6-4
6-5
6-6
Put the name of the
chapter on the front flap.
Proportions
and
Similarity
Reading and Writing As you read and study the chapter, use the Foldable to write down questions you
have about the concepts in each lesson. Leave room to record the answers to your questions.
Chapter 6 Proportions and Similarity 281
Proportions
• Write ratios.
• Use properties of proportions.
do artists use ratios?
Vocabulary
•
•
•
•
•
ratio
proportion
cross products
extremes
means
Stained-glass artist Louis Comfort Tiffany
used geometric shapes in his designs. In
a portion of Clematis Skylight shown at
the right, rectangular shapes are used as
the background for the flowers and vines.
Tiffany also used ratio and proportion in
the design of this piece.
WRITE RATIOS A ratio is a comparison of two quantities. The ratio of a to b
a
can be expressed as бЋЏбЋЏ, where b is not zero. This ratio can also be written as a:b.
b
Example 1 Write a Ratio
SOCCER The U.S. Census Bureau surveyed 8218 schools nationally about
their girls’ soccer programs. They found that 270,273 girls participated in a high
school soccer program in the 1999–2000 school year. Find the ratio of girl soccer
players per school to the nearest tenth.
Divide the number of girl soccer players by the number of schools.
number of girl soccer players
270,273
бЋЏбЋЏбЋЏбЋЏ П­ бЋЏбЋЏ or about 32.9
8,218
number of schools
A ratio in which the denominator is 1 is called a unit ratio.
The ratio for this survey was 32.9 girl soccer players for each school.
SOL/EOC Practice
Extended ratios can be used to compare three or more numbers. The expression
a:b:c means that the ratio of the first two numbers is a:b, the ratio of the last two
numbers is b:c, and the ratio of the first and last numbers is a:c.
Standardized Example 2 Extended Ratios in Triangles
Test Practice
Multiple-Choice Test Item
In a triangle, the ratio of the measures of three sides is 4:6:9, and its perimeter is
190 inches. Find the length of the longest side of the triangle.
A
10 in.
B
60 in.
C
90 in.
D
100 in.
Read the Test Item
You are asked to apply the ratio to the three sides of the triangle and the perimeter
to find the longest side.
282 Chapter 6 Proportions and Similarity
Christie’s Images
Solve the Test Item
Recall that equivalent fractions can be found by multiplying the numerator and the
Test-Taking Tip
Extended ratio problems
require a variable to be
the common factor among
the terms of the ratio. This
will enable you to write
an equation to solve the
problem.
2
3
x
x
2x
3x
denominator by the same number. So, 2 : 3 П­ бЋЏбЋЏ Рё бЋЏбЋЏ or бЋЏбЋЏ. Thus, we can rewrite
4 : 6 : 9 as 4x : 6x : 9x and use those measures for the sides of the triangle. Write an
equation to represent the perimeter of the triangle as the sum of the measures of
its sides.
4x П© 6x П© 9x П­ 190 Perimeter
9x
19x П­ 190 Combine like terms.
x П­ 10
4x
Divide each side by 19.
6x
Use this value of x to find the measures of the sides of the triangle.
4x П­ 4(10) or 40 inches
6x П­ 6(10) or 60 inches
9x П­ 9(10) or 90 inches
The longest side is 90 inches. The answer is C.
CHECK
Add the lengths of the sides to make sure that the perimeter is 190.
40 П© 60 П© 90 П­ 190 Я›
Study Tip
Reading
Mathematics
When a proportion is
written using colons, it is
read using the word to for
the colon. For example,
2:3 is read 2 to 3. The
means are the inside
numbers, and the
extremes are the outside
numbers.
USE PROPERTIES OF PROPORTIONS An equation stating that two ratios
are equal is called a proportion . Equivalent fractions set equal to each other form a
2
3
6
9
2
3
6
9
proportion. Since бЋЏбЋЏ and бЋЏбЋЏ are equivalent fractions, бЋЏбЋЏ П­ бЋЏбЋЏ is a proportion.
2
3
6
9
Every proportion has two cross products. The cross products in бЋЏбЋЏ П­ бЋЏбЋЏ are 2 times
9 and 3 times 6. The extremes of the proportion are 2 and 9. The means are 3 and 6.
2
6
бЋЏбЋЏ П­ бЋЏбЋЏ
3
9
extremes
2(9) П­ 3(6)
18 П­ 18
cross product of extremes
extremes
means
cross product of means
О†
О†
2:3 П­ 6:9
means
The product of the means equals the product of the extremes, so the cross products
are equal. Consider the general case.
a
c
бЋЏбЋЏ П­ бЋЏбЋЏ
b
d
a
c
(bd)бЋЏбЋЏ П­ (bd)бЋЏбЋЏ
b
d
da П­ bc
ad П­ bc
b
0, d
0
Multiply each side by the common denominator, bd.
Simplify.
Commutative Property
Property of Proportions
• Words
For any numbers a and c and any nonzero numbers b and d,
a
c
бЋЏбЋЏ П­ бЋЏбЋЏ if and only if ad П­ bc.
b
• Examples
d
4
12
бЋЏбЋЏ П­ бЋЏбЋЏ if and only if 4 Рё 15 П­ 5 Рё 12.
5
15
To solve a proportion means to find the value of the variable that makes the
proportion true.
www.geometryonline.com/extra_examples/sol
Lesson 6-1 Proportions
283
Example 3 Solve Proportions by Using Cross Products
Solve each proportion.
3x ПЄ 5
ПЄ13
b. бЋЏбЋЏ П­ бЋЏбЋЏ
x
3
a. бЋЏбЋЏ П­ бЋЏбЋЏ
75
5
3
x
бЋЏбЋЏ П­ бЋЏбЋЏ
5
75
3(75) П­ 5x
225 П­ 5x
45 П­ x
4
2
3x ПЄ 5
ПЄ13
бЋЏ бЋЏ П­ бЋЏбЋЏ
4
2
Original proportion
Original proportion
(3x ПЄ 5)2 П­ 4(ПЄ13) Cross products
Cross products
6x ПЄ 10 П­ ПЄ52
Multiply.
Divide each side by 5.
Simplify.
6x П­ ПЄ42
Add 10 to each side.
x П­ ПЄ7
Divide each side by 6.
Proportions can be used to solve problems involving two objects that are said to
be in proportion. This means that if you write ratios comparing the measures of all
parts of one object with the measures of comparable parts of the other object, a true
proportion would always exist.
Study Tip
Common
Misconception
The proportion shown in
Example 4 is not the only
correct proportion. There
are four equivalent
proportions:
c a
c
a
бЋЏбЋЏ П­ бЋЏбЋЏ, бЋЏбЋЏ П­ бЋЏбЋЏ,
d d
b
b
b
d
b
d
бЋЏбЋЏ П­ бЋЏбЋЏ, and бЋЏбЋЏ П­ бЋЏбЋЏ.
a
c
c
a
Example 4 Solve Problems Using Proportions
AVIATION A twinjet airplane has a length of 78 meters and a wingspan of
90 meters. A toy model is made in proportion to the real airplane. If the
wingspan of the toy is 36 centimeters, find the length of the toy.
Because the toy airplane and the real plane are in proportion, you can write a
proportion to show the relationship between their measures. Since both ratios
compare meters to centimeters, you need not convert all the lengths to the same
unit of measure.
plane’s length (m)
plane’s wingspan (m)
бЋЏбЋЏбЋЏ П­ бЋЏбЋЏбЋЏ
model’s length (cm)
model’s wingspan (cm)
All of these have identical
cross products.
78
90
бЋЏбЋЏ П­ бЋЏбЋЏ
x
36
Substitution
(78)(36) П­ x Рё 90 Cross products
2808 П­ 90x
31.2 П­ x
Multiply.
Divide each side by 90.
The length of the model would be 31.2 centimeters.
Concept Check
28
48
21
x
1. Explain how you would solve бЋЏбЋЏ П­ бЋЏбЋЏ.
2. OPEN ENDED Write two possible proportions having the extremes 5 and 8.
15
x
3
4
3. FIND THE ERROR Madeline and Suki are solving бЋЏбЋЏ П­ бЋЏбЋЏ.
Madeline
15
3
бЋЏбЋЏ = бЋЏбЋЏ
x
4
15 3
бЋЏбЋЏ = бЋЏбЋЏ
4
x
45 = 4x
11.25 = x
60 = 3x
20 = x
Who is correct? Explain your reasoning.
284 Chapter 6 Proportions and Similarity
Suki
Guided Practice
4. HOCKEY A hockey player scored 9 goals in 12 games. Find the ratio of
goals to games.
5. SCULPTURE A replica of The Thinker is
10 inches tall. A statue of The Thinker, located
in front of Grawemeyer Hall on the Belnap
Campus of the University of Louisville in
Kentucky, is 10 feet tall. What is the ratio of
the replica to the statue in Louisville?
Solve each proportion.
x
11
6. бЋЏбЋЏ П­ бЋЏбЋЏ
5
35
2.3
x
7. бЋЏбЋЏ П­ бЋЏбЋЏ
4
xПЄ2
4
8. бЋЏ
бЋЏ П­ бЋЏбЋЏ
3.7
2
5
9. The ratio of the measures of three sides of a triangle is 9 : 8 : 7, and its perimeter is
144 units. Find the measure of each side of the triangle.
10. The ratio of the measures of three angles of a triangle 5 : 7 : 8. Find the measure of
each angle of the triangle.
Standardized
Test Practice
11. GRID IN The scale on a map indicates that 1.5 centimeters represent 200 miles.
If the distance on the map between Norfolk, Virginia, and Atlanta, Georgia,
measures 2.4 centimeters, how many miles apart are the cities?
Practice and Apply
12. BASEBALL
to games.
A designated hitter made 8 hits in 10 games. Find the ratio of hits
For
Exercises
See
Examples
12–17,
23, 25
18–22
26, 27
28–35
1
13. SCHOOL There are 76 boys in a sophomore class of 165 students. Find the ratio
of boys to girls.
2
4
3
14. CURRENCY In a recent month, 208 South African rands were equivalent to
18 United States dollars. Find the ratio of rands to dollars.
Extra Practice
See page 764.
15. EDUCATION In the 2000–2001 school year, Arizona State University had
44,125 students and 1747 full-time faculty members. What was the ratio of the
students to each teacher rounded to the nearest tenth?
16. Use the number line at the right to
determine the ratio of AC to BH.
A B C D E F G H
I
0
80
20
40
60
17. A cable that is 42 feet long is divided into lengths in the ratio of 3:4. What are
the two lengths into which the cable is divided?
Find the measures of the angles of each triangle.
18. The ratio of the measures of the three angles is 2 : 5 : 3.
19. The ratio of the measures of the three angles is 6 : 9 : 10.
Find the measures of the sides of each triangle.
20. The ratio of the measures of three sides of a triangle is 8 : 7 : 5. Its perimeter is
240 feet.
21. The ratio of the measures of the sides of a triangle is 3 : 4 : 5. Its perimeter is
72 inches.
1 1 1
22. The ratio of the measures of three sides of a triangle are бЋЏбЋЏ : бЋЏбЋЏ : бЋЏбЋЏ, and its
2 3 5
perimeter is 6.2 centimeters. Find the measure of each side of the triangle.
Lesson 6-1 Proportions
285
Courtesy University of Louisville
LITERATURE For Exercises 23 and 24, use the following information.
Throughout Lewis Carroll’s book, Alice’s Adventures in Wonderland, Alice’s size
changes. Her normal height is about 50 inches tall. She comes across a door, about
15 inches high, that leads to a garden. Alice’s height changes to 10 inches so she can
visit the garden.
23. Find the ratio of the height of the door to Alice’s height in Wonderland.
24. How tall would the door have been in Alice’s normal world?
25. ENTERTAINMENT Before actual construction of the Great Moments with
Mr. Lincoln exhibit, Walt Disney and his design company built models that
were in proportion to the displays they planned to build. What is the ratio
of the height of the model of Mr. Lincoln compared to his actual height?
Entertainment
In the model, Lincoln
is 8 inches tall. In the
theater, Lincoln is
6 feet 4 inches tall (his
actual adult height).
Source: Disney
ICE CREAM For Exercises 26 and 27, use the following information.
There were approximately 255,082,000 people in the United States in a recent
year. According to figures from the United States Census, they consumed about
4,183,344,800 pounds of ice cream that year.
26. If there were 276,000 people in the city of Raleigh, North Carolina, about how
much ice cream might they have been expected to consume?
27. Find the approximate consumption of ice cream per person.
Online Research Data Update Use the Internet or other resource to
find the population of your community. Determine how much ice cream you
could expect to be consumed each year in your community. Visit
www.geometryonline.com/data_update to learn more.
ALGEBRA
Solve each proportion.
3
x
28. бЋЏбЋЏ П­ бЋЏбЋЏ
8
5
2x ПЄ 13
ПЄ4
32. бЋЏбЋЏ П­ бЋЏбЋЏ
28
7
a
1
29. бЋЏбЋЏ П­ бЋЏбЋЏ
5.18
4
4x П© 3
5
33. бЋЏбЋЏ П­ бЋЏбЋЏ
12
4
3x
48
30. бЋЏбЋЏ П­ бЋЏбЋЏ
13
26
31. бЋЏбЋЏ П­ бЋЏбЋЏ
23
92
bП©1
5
34. бЋЏбЋЏ П­ бЋЏбЋЏ
bПЄ1
6
49
7x
ПЄ2
3x ПЄ 1
35. бЋЏбЋЏ П­ бЋЏбЋЏ
xП©2
2
PHOTOGRAPHY For Exercises 36 and 37, use the following information.
JosГ© reduced a photograph that is 21.3 centimeters by 27.5 centimeters so that it
would fit in a 10-centimeter by 10-centimeter area.
36. Find the maximum dimensions of the reduced photograph.
37. What percent of the original length is the length of the reduced photograph?
38. CRITICAL THINKING The ratios of the lengths of the sides of three polygons are
given below. Make a conjecture about identifying each type of polygon.
a. 2 : 2 : 3
b. 3 : 3 : 3 : 3
c. 4 : 5 : 4 : 5
39. WRITING IN MATH
Answer the question that was posed at the beginning of
the lesson.
How do artists use ratios?
Include the following in your answer:
• four rectangles from the photo that appear to be in proportion, and
• an estimate in inches of the ratio of the width of the skylight to the length
of the skylight given that the dimensions of the rectangle in the bottom left
corner are approximately 3.5 inches by 5.5 inches.
286 Chapter 6 Proportions and Similarity
Walt Disney Co.
Standardized
Test Practice
40. SHORT RESPONSE In a golden rectangle, the ratio of the length of the
rectangle to its width is approximately 1.618:1. Suppose a golden rectangle has
a length of 12 centimeters. What is its width to the nearest tenth?
SOL/EOC Practice
41. ALGEBRA A breakfast cereal contains wheat, rice, and oats in the ratio 3:1:2. If
the manufacturer makes a mixture using 120 pounds of oats, how many pounds
of wheat will be used?
A 60 lb
B 80 lb
C 120 lb
D 180 lb
Maintain Your Skills
Mixed Review
In the figure, а·†
Sа·†
O is a median of б­ќSLN, а·†
Oа·†
SРҐа·†
NP
а·†, mР„1 П­ 3x ПЄ 50,
and mР„2 П­ x П© 30. Determine whether each statement is always,
sometimes, or never true. (Lesson 5-5)
42. LS Пѕ SN
43. SN ПЅ OP
44. x П­ 45
S
1 2
L
O
N
P
Find the range for the measure of the third side of a triangle given the measures
of two sides. (Lesson 5-4)
45. 16 and 31
46. 26 and 40
47. 11 and 23
48. COORDINATE GEOMETRY Given б­ќSTU with vertices S(0, 5), T(0, 0), and
U(ПЄ2, 0) and б­ќXYZ with vertices X(4, 8), Y(4, 3), and Z(6, 3), show that
б­ќSTU РҐ б­ќXYZ. (Lesson 4-4)
Graph the line that satisfies each condition. (Lesson 3-3)
3
5
49. m П­ бЋЏбЋЏ and contains P(ПЄ3, ПЄ4)
50. contains A(5, 3) and B(ПЄ1, 8)
51. parallel to бџ­бџ®бџ¬
JK with J(ПЄ1, 5) and K(4, 3) and contains E(2, 2)
52. contains S(8, 1) and is perpendicular to бџ­бџ®бџ¬
QR with Q(6, 2) and R(ПЄ4, ПЄ6)
53. MAPS On a U.S. map, there is a scale that lists kilometers on the top and miles
on the bottom.
kilometers
0
miles
0
20
40
50
60
80
100
31
62
Suppose а·†
AB
а·† and C
а·†D
а·† are segments on this map. If AB П­ 100 kilometers and
CD П­ 62 miles, is а·†
AB
CD
а·†РҐа·†
а·†? Explain. (Lesson 2-7)
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Find the distance between each pair of points to the nearest
tenth. (To review the Distance Formula, see Lesson 1-3.)
54. A(12, 3), B(ПЄ8, 3)
55. C(0, 0), D(5, 12)
ПЄ1
2
56. EО‚бЋЏбЋЏ, ПЄ1Оѓ, FО‚2, бЋЏбЋЏОѓ
4
5
www.geometryonline.com/self_check_quiz /sol
57. GО‚3, бЋЏбЋЏОѓ, HО‚4, ПЄбЋЏбЋЏОѓ
3
7
2
7
Lesson 6-1 Proportions
287
A Follow-Up of Lesson 6-1
Fibonacci Sequence and Ratios
The Fibonacci sequence is a set of numbers that begins with 1 as its first and second
terms. Each successive term is the sum of the two numbers before it. This sequence
continues on indefinitely.
term
1
2
3
4
5
6
7
Fibonacci number
1
1
2
3
5
8
13
в†ђ
в†ђ
в†ђ
в†ђ
в†ђ
1П©1
1П©2
2П©3
3П©5
5П©8
Example
Use a spreadsheet to create twenty terms of the Fibonacci sequence. Then compare
each term with its preceding term.
Step 1 Enter the column headings in
rows 1 and 2.
Step 2 Enter 1 into cell A3. Then insert the
formula П­A3 П© 1 in cell A4. Copy
this formula down the column.
This will automatically calculate
the number of the term.
Step 3 In column B, we will record the
Fibonacci numbers. Enter 1 in
cells B3 and B4 since you do not
have two previous terms to add.
Then insert the formula П­B3 П© B4
in cell B5. Copy this formula
down the column.
term
Fibonacci
number
n
1
2
3
4
5
6
7
F(n)
1
1
2
3
5
8
13
ratio
F(n+1)/F(n)
1
1
2
1.5
1.666666667
1.6
1.625
Step 4 In column C, we will find the
ratio of each term to its preceding
term. Enter 1 in cell C3 since there is no preceding term. Then enter П­B4/B3 in
cell C4. Copy this formula down the column.
Exercises
1.
2.
3.
4.
What happens to the Fibonacci number as the number of the term increases?
What pattern of odd and even numbers do you notice in the Fibonacci sequence?
As the number of terms gets greater, what pattern do you notice in the ratio column?
Extend the spreadsheet to calculate fifty terms of the Fibonacci sequence. Describe
any differences in the patterns you described in Exercises 1–3.
The rectangle that most humans perceive to be pleasing to the eye has a width to
length ratio of about 1:1.618. This is called the golden ratio, and the rectangle is called
the golden rectangle. This type of rectangle is visible in nature and architecture. The
Fibonacci sequence occurs in nature in patterns that are also pleasing to the human
eye, such as in sunflowers, pineapples, and tree branch structure.
5. MAKE A CONJECTURE
How might the Fibonacci sequence relate to the golden ratio?
288 Chapter 6 Proportions and Similarity
288 Chapter 6
NETS 7784
Similar Polygons
Virginia SOL Standard G.14a The student will use proportional reasoning to solve practical problems,
given similar geometric objects;
• Identify similar figures.
Vocabulary
• similar polygons
• scale factor
В©2002 Cordon Art B.V., Baarn, Holland. All rights reserved.
• Solve problems involving scale factors.
do artists use
geometric patterns?
M.C. Escher (1898–1972) was a Dutch
graphic artist known for drawing
impossible structures, spatial illusions,
and repeating interlocking geometric
patterns. The image at the right is a print
of Escher’s Circle Limit IV, which is
actually a woodcutting. It includes winged
images that have the same shape, but are
different in size. Also note that there are not only
similar dark images but also similar light images.
Circle Limit IV,
M.C. Escher (1960)
IDENTIFY SIMILAR FIGURES When polygons have the same shape but may
be different in size, they are called similar polygons.
Similar Polygons
• Words
Two polygons are similar if and only if their corresponding angles are
congruent and the measures of their corresponding sides are
proportional.
Similarity and
Congruence
• Symbol ϳ is read is similar to
• Example
D
10 m
H
5m
A
8m
5m
B
E
2.5 m
F
6m
4m
3m G
C
The order of the vertices in a similarity statement is important. It identifies the
corresponding angles and the corresponding sides.
в†ђ
в†ђ
similarity statement
в†ђ
If two polygons are
congruent, they are also
similar. All of the
corresponding angles are
congruent, and the lengths
of the corresponding sides
have a ratio of 1:1.
в†ђ
Study Tip
в†ђ
в†ђ
в†ђ
в†ђ
ABCD Пі EFGH
congruent angles
corresponding sides
Р„A РҐ Р„E
Р„B РҐ Р„F
Р„C РҐ Р„G
Р„D РҐ Р„H
CD
AB
BC
DA
бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ
GH
EF
FG
HE
Like congruent polygons, similar polygons may be repositioned so that
corresponding parts are easy to identify.
Lesson 6-2 Similar Polygons 289
Art Resource, NY
Example 1 Similar Polygons
Study Tip
Identifying
Corresponding
Parts
Using different colors to
circle letters of congruent
angles may help you
identify corresponding
parts.
Determine whether each pair of figures is similar. Justify your answer.
a.
B
12
A
F
6
30Лљ
6в€љ3
4.5 в€љ 3
4.5
C
30Лљ
9
E
D
All right angles are congruent, so Р„C РҐ Р„F.
Since mР„A П­ mР„D, Р„A РҐ Р„D.
By the Third Angle Theorem, Р„B РҐ Р„E.
Thus, all corresponding angles are congruent.
Now determine whether corresponding sides are proportional.
Sides opposite 90В° angle
Sides opposite 30В° angle
Sides opposite 60В° angle
AB
12
бЋЏбЋЏ П­ бЋЏбЋЏ or 1.3
а·†
DE
9
6
BC
бЋЏбЋЏ П­ бЋЏбЋЏ or 1.3
а·†
4.5
EF
6Н™3а·†
AC
бЋЏбЋЏ П­ бЋЏ or 1.3
а·†
DF
4.5Н™3
а·†
The ratios of the measures of the corresponding sides are equal, and the
corresponding angles are congruent, so б­ќABC Пі б­ќDEF.
Study Tip
Common
Misconception
When two figures have
vertices that are in
alphabetical order, this
does not mean that the
corresponding vertices in
the similarity statement will
follow alphabetical order.
b. A
7
6
D
B
E
5
6
7
6
5
H
C
F
6
G
Both rectangles have all right angles and right angles are congruent.
AB
7
BC
6
AB
бЋЏбЋЏ П­ бЋЏбЋЏ and бЋЏбЋЏ П­ бЋЏбЋЏ, but бЋЏбЋЏ
EF
6
FG
5
EF
BC
7
бЋЏбЋЏ because бЋЏбЋЏ
FG
6
6
бЋЏбЋЏ. The rectangles are not similar.
5
SCALE FACTORS When you compare the lengths of corresponding sides of
similar figures, you usually get a numerical ratio. This ratio is called the scale factor
for the two figures. Scale factors are often given for models of real-life objects.
Example 2 Scale Factor
MOVIES Some special effects in movies are created using miniature models. In
a recent movie, a model sports-utility vehicle (SUV) 22 inches long was created
2
to look like a real 14бЋЏбЋЏ -foot SUV. What is the scale factor of the model compared
3
to the real SUV?
Before finding the scale factor you must make sure that both measurements use
the same unit of measure.
2
3
length of model
22 inches
бЋЏбЋЏбЋЏ П­ бЋЏбЋЏ
176 inches
length of real SUV
14бЋЏбЋЏ(12) П­ 176 inches
1
8
П­ бЋЏбЋЏ
1
8
1
8
The ratio comparing the two lengths is бЋЏбЋЏ or 1: 8. The scale factor is бЋЏбЋЏ, which means
1
8
that the model is бЋЏбЋЏ the length of the real SUV.
290
Chapter 6 Proportions and Similarity
When finding the scale factor for two similar polygons, the scale factor will
depend on the order of comparison.
C
12 cm
D
10 cm
6 cm
H
G
6 cm
5 cm
3 cm
A
E 3.5 cm F
B
7 cm
• The scale factor of quadrilateral ABCD to quadrilateral EFGH is 2.
1
2
• The scale factor of quadrilateral EFGH to quadrilateral ABCD is ᎏᎏ.
Example 3 Proportional Parts and Scale Factor
R
The two polygons are similar.
a. Write a similarity statement.
Then find x, y, and UT.
xm
S
V
Use the congruent angles to write the
corresponding vertices in order.
polygon RSTUV Пі polygon ABCDE
ST
бЋЏбЋЏ П­
BC
18
бЋЏбЋЏ П­
4
VR
бЋЏбЋЏ
EA
x
бЋЏбЋЏ
3
54 П­ 4x
13.5 П­ x
T
(y П© 2) m
ST П­ 18, BC П­ 4
VR П­ x, EA П­ 3
4m
C
ST
UT
бЋЏбЋЏ П­ бЋЏбЋЏ
BC
DC
yП©2
18
бЋЏбЋЏ П­ бЋЏбЋЏ
5
4
Similarity proportion
E
U
To find y:
18(3) П­ 4(x) Cross products
3m
B
Now write proportions to find x and y.
To find x:
A
18 m
5m
D
Similarity proportion
ST П­ 18, BC П­ 4
UT П­ y П© 2, EA П­ 3
18(5) П­ 4(y П© 2) Cross products
Multiply.
90 П­ 4y П© 8
Multiply.
Divide each side by 4.
82 П­ 4y
Subtract 8 from each side.
20.5 П­ y
Divide each side by 4.
UT П­ y П© 2, so UT П­ 20.5 П© 2 or 22.5.
Study Tip
Checking
Solutions
To verify the scale factor,
find the ratio of two other
corresponding sides.
b. Find the scale factor of polygon RSTUV to polygon ABCDE.
The scale factor is the ratio of the lengths of any two corresponding sides.
18
9
ST
бЋЏ П­ бЋЏбЋЏ or бЋЏбЋЏ
4
2
BC
You can use scale factors to produce similar figures.
Example 4 Enlargement of a Figure
2
Triangle ABC is similar to б­ќXYZ with a scale factor of бЋЏбЋЏ. If the lengths of the
3
sides of б­ќABC are 6, 8, and 10 inches, what are the lengths of the sides of б­ќXYZ?
Write proportions for finding side measures.
б­ќABC в†’ 6
2
бЋЏбЋЏ П­ бЋЏбЋЏ
3
б­ќXYZ в†’ x
б­ќABC в†’ бЋЏ8бЋЏ П­ бЋЏ2бЋЏ
3
б­ќXYZ в†’ y
б­ќABC в†’ 10
2
бЋЏбЋЏ П­ бЋЏбЋЏ
3
б­ќXYZ в†’ z
18 П­ 2x
24 П­ 2y
30 П­ 2z
9П­x
12 П­ y
15 П­ z
The lengths of the sides of б­ќXYZ are 9, 12, and 15 inches.
www.geometryonline.com/extra_examples/sol
Lesson 6-2 Similar Polygons 291
Example 5 Scale Factors on Maps
MAPS The scale on the map of New
Mexico is 2 centimeters П­ 160 miles. The
distance on the map across New Mexico
from east to west through Albuquerque is
4.1 centimeters. How long would it take to
drive across New Mexico if you drove at
an average of 60 miles per hour?
Albuquerque
Roswell
Las Cruces
Explore
Every 2 centimeters represents
160 miles. The distance across
the map is 4.1 centimeters.
Plan
Create a proportion relating the measurements to the scale to find the
distance in miles. Then use the formula d П­ rt to find the time.
Solve
centimeters в†’
miles в†’
2
4.1
бЋЏбЋЏ П­ бЋЏбЋЏ
160
x
в†ђ centimeters
в†ђ miles
2x П­ 656
Cross products
x П­ 328
Study Tip
Divide each side by 2.
The distance across New Mexico is approximately 328 miles.
Units of Time
d П­ rt
Remember that there are
60 minutes in an hour.
328 П­ 60t d П­ 328 and r П­ 60
328
60
When rewriting бЋЏбЋЏ as a
328
бЋЏбЋЏ П­ t
60
7
5бЋЏбЋЏ П­ t
15
mixed number, you could
28
also write 5бЋЏбЋЏ, which
60
means 5 hours 28
minutes.
Divide each side by 60.
Simplify.
7
15
It would take 5бЋЏбЋЏ hours or about 5 hours and 28 minutes to drive
across New Mexico at an average of 60 miles per hour.
Examine Reexamine the scale. If 2 centimeters П­ 160 miles, then 4 centimeters П­
320 miles. The map is about 4 centimeters wide, so the distance across
New Mexico is about 320 miles. The answer is about 5.5 hours and at
60 miles per hour, the trip would be 330 miles. The two distances are
close estimates, so the answer is reasonable.
Concept Check
1. FIND THE ERROR Roberto and Garrett have calculated their scale factor for
two similar triangles.
Roberto
Garrett
AB
8
бЋЏбЋЏ П­ бЋЏбЋЏ
PQ
10
4
П­ бЋЏбЋЏ
5
10
PQ
бЋЏбЋЏ П­ бЋЏбЋЏ
8
AB
5
П­ бЋЏбЋЏ
4
Who is correct? Explain your reasoning.
292 Chapter 6 Proportions and Similarity
B
P
8 in.
A
C
10 in.
R
Q
2. Find a counterexample for the statement All rectangles are similar.
3. OPEN ENDED Explain whether two polygons that are congruent are also
similar. Then explain whether two polygons that are similar are also congruent.
Guided Practice
Determine whether each pair of figures is similar. Justify your answer.
4.
5.
G
F
60Лљ
P
60Лљ
3
Q
R
B
4
C
3
7
G
9
2
9
2
3
A
60Лљ
6
E
D
4
H
6
H
I
Each pair of polygons is similar. Write a similarity statement, and find x, the
measure(s) of the indicated side(s), and the scale factor.
F
7. F
E, E
H, and G
F
6. D
а·†а·†
а·†а·†
а·†а·†
а·†а·†
F
A
G
B
D
21
16
xП©5
C
14
x
C
B
27
F
18
H
A 10 D
E
xПЄ3
E
8. A rectangle with length 60 centimeters and height 40 centimeters is reduced so
that the new rectangle is similar to the original and the scale factor is бЋЏ1бЋЏ. Find the
4
length and height of the new rectangle.
9. A triangle has side lengths of 3 meters, 5 meters, and 4 meters. The triangle is
enlarged so that the larger triangle is similar to the original and the scale factor
is 5. Find the perimeter of the larger triangle.
Application
10. MAPS Refer to Example 5 on page 292. Draw the state of New Mexico using a
scale of 2 centimeters П­ 100 miles. Is your drawing similar to the one in
Example 4? Explain how you know.
Practice and Apply
For
Exercises
See
Examples
11–14
15–20,
34–39
21–23
24–26
1
2, 3
Determine whether each pair of figures is similar. Justify your answer.
C
11.
12.
X
4
D
120Лљ
130Лљ E
3
Z
10
2
1
50Лљ 50Лљ
4
5
Y
10
W
F
Extra Practice
See page 765.
6 5
60Лљ 60Лљ
B 120Лљ
A 130Лљ
13. A
14.
E
4 36.9Лљ 5
3.5
M
1
53
52.1Лљ
B
5.7
2
C
53.1Лљ
3
B
F
37.9Лљ
4.5
D
D
4
P
3
C
8
2
N
10 3
Lesson 6-2 Similar Polygons 293
15. ARCHITECTURE The replica of the Eiffel Tower at an amusement park is
2
350бЋЏбЋЏ feet tall. The actual Eiffel Tower is 1052 feet tall. What is the scale factor
3
comparing the amusement park tower to the actual tower?
16. PHOTOCOPYING Mr. Richardson walked to a copier in his office, made
a copy of his proposal, and sent the original to one of his customers. When
Mr. Richardson looked at his copy before filing it, he saw that the copy had
been made at an 80% reduction. He needs his filing copy to be the same size
as the original. What enlargement scale factor must he use on the first copy to
make a second copy the same size as the original?
Each pair of polygons is similar. Write a similarity statement, and find x, the
measures of the indicated sides, and the scale factor.
B and C
D
18. A
C and C
E
17. A
а·†а·†
а·†а·†
а·†а·†
а·†а·†
B
D
F
8
E
6
C
x–1
x+1
E
x+7
D
A
H
5
G
C
4
B
A
19. B
C and E
а·†а·†
а·†D
а·†
12 – x
20. G
F and E
а·†а·†
а·†G
а·†
A
10
B
x+2
G
S
20.7
6.25
E
x–1
R
D
43Лљ
10
27Лљ 110Лљ
T
15
27Лљ
E
x
110Лљ
F
11.25
C
PHOTOGRAPHY For Exercises 21–23, use the following information.
5
A picture is enlarged by a scale factor of бЋЏбЋЏ and then enlarged again by the same factor.
4
21. If the original picture was 2.5 inches by 4 inches, what were its dimensions after
both enlargements?
22. Write an equation describing the enlargement process.
23. By what scale factor was the original picture enlarged?
SPORTS Make a scale drawing of each playing field using the given scale.
24. Use the information about the soccer field in Crew Stadium. Use the scale
1 millimeter П­ 1 meter.
1
4
25. A basketball court is 84 feet by 50 feet. Use the scale бЋЏбЋЏ inch П­ 4 feet.
1
8
26. A tennis court is 36 feet by 78 feet. Use the scale бЋЏбЋЏ inch П­ 1 foot.
Sports
Crew Stadium in
Columbus, Ohio, was
specifically built for Major
League Soccer. The
dimensions of the field
are about 69 meters by
105 meters.
Source: www.MLSnet.com
Determine whether each statement is always, sometimes, or never true.
27. Two congruent triangles are similar.
28. Two squares are similar.
29. A triangle is similar to a quadrilateral.
30. Two isosceles triangles are similar.
31. Two rectangles are similar.
32. Two obtuse triangles are similar.
33. Two equilateral triangles are similar.
294 Chapter 6 Proportions and Similarity
Joe Giblin/Columbus Crew/MLS
Each pair of polygons is similar. Find x and y. Round to the nearest hundredth if
necessary.
R
I
34. H
35. K
(y П© 30)Лљ
87Лљ
6
60Лљ
G
36.
N
M
98Лљ
J
L
xП©2
37.
15
8
E
38.
12
16
5
10
xПЄ3
H
39. R
12
yП©4
2x
8
yП©1
G
C
S
10
12
D
yПЄ3
B
8
80Лљ x
Лљ
L
J
F
A
Q
30Лљ
O
(x ПЄ 4)Лљ
4
12
yЛљ
15
yП©3
T
x
49
W
20
29
20
S
V
21
U
For Exercises 40–47, use the following information to find each measure.
Polygon ABCD Пі polygon AEFG, mР„AGF П­ 108, GF П­ 14, AD П­ 12, DG П­ 4.5,
EF П­ 8, and AB П­ 26.
40.
41.
42.
43.
44.
45.
46.
47.
scale factor of trapezoid ABCD to trapezoid AEFG
26
E
B
AG
A
8
DC
108Лљ
12 G
F
mР„ADC
4.5 14
BC
C
D
perimeter of trapezoid ABCD
perimeter of trapezoid AEFG
ratio of the perimeter of polygon ABCD to the perimeter of polygon AEFG
48. Determine which of the following right triangles are similar. Justify your
answer.
G
E
A
37Лљ
20
43.8
5
3 53Лљ
D
8
10
H
6
27Лљ
39
B
C
4
1.25
J
1
L
N
K
53Лљ
0.75
F
5
M
P
67Лљ
I
12.5 R
67Лљ
13
30
32.5
12
O
S
COORDINATE GEOMETRY Graph the given points. Draw polygon ABCD and а·†
Mа·†
N.
Find the coordinates for vertices L and P such that ABCD ПіNLPM.
49. A(2, 0), B(4, 4), C(0, 4), D(ПЄ2, 0); M(4, 0), N(12, 0)
50. A(ПЄ7, 1), B(2, 5), C(7, 0), D(ПЄ2, ПЄ4); M(ПЄ3, 1), NО‚ПЄбЋЏбЋЏ, бЋЏбЋЏОѓ
11 7
2 2
Lesson 6-2 Similar Polygons 295
CONSTRUCTION For Exercises 51 and 52,
use the following information.
A floor plan is given for the first floor of a
new house. One inch represents 24 feet.
Use the information in the plan to find
the dimensions.
5
in.
8
3
in.
4
1 4 in.
1
Living
Room
Deck
Kitchen
51. living room
52. deck
3
in.
8
Master
Suite
Dining
Room
CRITICAL THINKING For Exercises 53–55, use the following information.
The area A of a rectangle is the product of its length бђ‰ and width w. Rectangle
ABCD is similar to rectangle WXYZ with sides in a ratio of 4:1.
53. What is the ratio of the areas of the two rectangles?
54. Suppose the dimension of each rectangle is tripled. What is the new ratio of the
sides of the rectangles?
55. What is the ratio of the areas of these larger rectangles?
STATISTICS For Exercises 56–58,
refer to the graphic, which uses
rectangles to represent percents.
56. Are the rectangles representing
36% and 18% similar? Explain.
USA TODAY SnapshotsВ®
Workplace manners declining
A survey asked workers whether they thought the level
of professional courtesy in the workplace had increased
or decreased in the past five years. Their responses:
57. What is the ratio of the areas of
the rectangles representing 36%
and 18% if area П­ lengthП«
width? Compare the ratio of the
areas to the ratio of the percents.
58. Use the graph to make a
conjecture about the overall
changes in the level of
professional courtesy in the
workplace in the past five years.
44%
Decreased
Increased
36%
Not changed
18%
Don’t know/no answer
2%
Source: OfficeTeam poll of 525 adults Feb. 7-13.
Margin of error: +/–4.3 percentage points.
By Sam Ward, USA TODAY
CRITICAL THINKING For Exercises 59 and 60,
б­ќABC Пі б­ќDEF.
A
59. Show that the perimeters of б­ќABC and
b
б­ќDEF have the same ratio as their
C
corresponding sides.
60. If 6 units are added to the lengths of
each side, are the new triangles similar?
D
c
a
B
3b
F
3c
3a
61. WRITING IN MATH
E
Answer the question that was posed at the beginning of
the lesson.
How do artists use geometric patterns?
Include the following in your answer:
• why Escher called the picture Circle Limit IV, and
• how one of the light objects and one of the dark objects compare in size.
296 Chapter 6 Proportions and Similarity
SOL/EOC Practice
Standardized
Test Practice
62. In a history class with 32 students, the ratio of girls to boys is 5 to 3. How many
more girls are there than boys?
A 2
B 8
C 12
D 15
63. ALGEBRA Find x.
A 4.2
C 5.6
B
D
2.8 m
51Лљ 85Лљ
4.65
8.4
xm
51Лљ
3.1 m
4m
9.3 m
12 m
44Лљ
Extending
the Lesson
Scale factors can be used to produce similar figures. The resulting figure is an
enlargement or reduction of the original figure depending on the scale factor.
Triangle ABC has vertices A(0, 0), B(8, 0), and C(2, 7). Suppose the coordinates of
each vertex are multiplied by 2 to create the similar triangle A’B’C’.
64. Find the coordinates of the vertices of ᭝A’B’C’.
65. Graph ᭝ABC and ᭝A’B’C’.
66. Use the Distance Formula to find the measures of the sides of each triangle.
67. Find the ratios of the sides that appear to correspond.
68. How could you use slope to determine if angles are congruent?
69. Is ᭝ABC ϳ ᭝A’B’C’? Explain your reasoning.
Maintain Your Skills
Mixed Review
Solve each proportion. (Lesson 6-1)
cПЄ2
5
71. бЋЏбЋЏ П­ бЋЏбЋЏ
b
2
70. бЋЏбЋЏ П­ бЋЏбЋЏ
7.8
cП©3
3
2
ПЄ4
72. бЋЏбЋЏ П­ бЋЏбЋЏ
4y П© 5
4
Use the figure to write an inequality relating
each pair of angle or segment measures. (Lesson 5-5)
73. OC, AO
74. mР„AOD, mР„AOB
75. mР„ABD, mР„ADB
Find x. (Lesson 4-2)
76.
xЛљ
52Лљ
y
B
10
68Лљ
C
40Лљ
10
A
O
9.3
10.2
D
77.
78.
57Лљ
40Лљ
xЛљ
35Лљ
32Лљ
25Лљ
xЛљ
79. Suppose two parallel lines are cut by a transversal and Р„1 and Р„2 are alternate
interior angles. Find mР„1 and mР„2 if mР„1 П­ 10x ПЄ 9 and mР„2 П­ 9x П© 3.
(Lesson 3-2)
Getting Ready for
the Next Lesson
CD
BD
PREREQUISITE SKILL In the figure, A
AC
а·†B
а·†аїЈа·†
а·†, а·†
а·†аїЈа·†
а·†, and mР„4 П­ 118. Find the
measure of each angle. (To review angles and parallel lines, see Lesson 3-2.)
80. Р„1
81. Р„2
A 1
3
2
82. Р„3
83. Р„5
4 C
84. Р„ABD
85. Р„6
B 5
86. Р„7
87. Р„8
6
7
8 D
www.geometryonline.com/self_check_quiz /sol
Lesson 6-2 Similar Polygons 297
Similar Triangles
Virginia SOL Standard G.5a The student will investigate and identify … similarity relationships between triangles;
Standard G.5b The student will prove two triangles are … similar, given information in the form of a figure or
statement, using algebraic and … deductive proofs.
• Identify similar triangles.
• Use similar triangles to solve problems.
do engineers use geometry?
The Eiffel Tower was built in Paris for the 1889 world exhibition by
Gustave Eiffel. Eiffel (1832–1923) was a French engineer who specialized
in revolutionary steel constructions. He used thousands of triangles,
some the same shape but different in size, to build the Eiffel Tower
because triangular shapes result in rigid construction.
IDENTIFY SIMILAR TRIANGLES In Chapter 4, you learned several tests to
determine whether two triangles are congruent. There are also tests to determine
whether two triangles are similar.
Similar Triangles
Collect Data
Eiffel Tower
The Eiffel Tower weighs
7000 tons, but the pressure
per square inch it applies
on the ground is only
equivalent to that of a chair
with a person seated in it.
Source: www.eiffel-tower.com
• Draw ᭝DEF with mЄD ϭ 35, mЄF ϭ 80, and DF ϭ 4 centimeters.
• Draw ᭝RST with mЄT ϭ 35, mЄS ϭ 80, and ST ϭ 7 centimeters.
ED
RS
RT
• Measure ෆ
EFа·†, а·†
а·†, а·†
а·†, and а·†
а·†.
FD EF
ED
• Calculate the ratios ᎏᎏ, ᎏᎏ, and ᎏᎏ.
ST R S
RT
Analyze the Data
1. What can you conclude about all of the ratios?
2. Repeat the activity with two more triangles with the same angle measures,
but different side measures. Then repeat the activity with a third pair of
triangles. Are all of the triangles similar? Explain.
3. What are the minimum requirements for two triangles to be similar?
The previous activity leads to the following postulate.
Postulate 6.1
Angle-Angle (AA) Similarity
If the two angles of
one triangle are congruent to two angles of another
triangle, then the triangles are similar.
Example: Р„P РҐ Р„T and Р„Q РҐ Р„S,
so б­ќPQR Пі б­ќTSU.
Q
P
S
R
T
U
You can use the AA Similarity Postulate to prove two theorems that also verify
triangle similarity.
298 Chapter 6 Proportions and Similarity
Jeremy Walker/Getty Images
Theorems
6.1 Side-Side-Side (SSS) Similarity
If the measures
of the corresponding sides of two triangles
are proportional, then the triangles are
similar.
6.2
SU
a
c
cx
T
Side-Angle-Side (SAS) Similarity If the
measures of two sides of a triangle are
proportional to the measures of two
corresponding sides of another triangle
and the included angles are congruent,
then the triangles are similar.
bx
R
UT
PQ
QR
Example: бЋЏбЋЏ П­ бЋЏбЋЏ and
ST
SU
S
b
ax
P
RP
PQ
QR
Example: бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ, so б­ќPQR Пі б­ќTSU.
ST
Q
Q
a
U
S
c
ax
P
cx
R
T
U
Р„Q РҐ Р„S, so б­ќPQR Пі б­ќTSU.
You will prove Theorem 6.2 in Exercise 34.
Proof
Theorem 6.1
RP
PQ
QR
Given: бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ
AB
BC
Q
B
CA
5
Prove: б­ќBAC Пі б­ќQPR
D 2 3 E
Locate D on а·†
AB
DB
PQ
а·† so that а·†
а·†РҐа·†
а·† and
аїЈ
AC
DE
draw а·†
DE
а·† so that а·†
а·† а·†
а·†.
1
PQ
AB
QR
BC
R
4
A
Paragraph Proof:
P
C
RP
CA
Since а·†
DB
PQ
а·†РҐа·†
а·†, DB П­ PQ. бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ
DB
AB
QR
BC
RP
CA
becomes бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ. Since а·†
DE
C, Р„2 РҐ Р„1 and Р„3 РҐ Р„4.
а·†аїЈA
а·†а·†
By AA Similarity, б­ќBDE Пі б­ќBAC.
DB
AB
BE
BC
ED
CA
By the definition of similar polygons, бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ. By substitution,
BE
RP
ED
QR
бЋЏбЋЏ П­ бЋЏбЋЏ and бЋЏбЋЏ П­ бЋЏбЋЏ. This means that QR П­ BE and RP П­ ED or
BC
CA
CA
BC
BE
ED
DB
PQ
Q
а·†R
а·†РҐа·†
а·† and R
а·†P
а·†РҐа·†
а·†. With these congruences and а·†
а·†РҐа·†
а·†, б­ќBDE РҐ б­ќQPR
by SSS. By CPCTC, Р„B РҐ Р„Q and Р„2 РҐ Р„P. But Р„2 РҐ Р„A, so Р„A РҐ Р„P.
By AA Similarity, б­ќBAC Пі б­ќQPR.
Study Tip
Overlapping
Triangles
When two triangles
overlap, draw them
separately so the
corresponding parts are
in the same position on
the paper. Then write
the corresponding
angles and sides.
Example 1 Determine Whether Triangles Are Similar
In the figure, F
EG
а·†G
а·†РҐа·†
а·†, BE П­ 15, CF П­ 20, AE П­ 9,
and DF П­ 12. Determine which triangles in the
figure are similar.
Triangle FGE is an isosceles triangle. So, Р„GFE РҐ Р„GEF.
If the measures of the corresponding sides that
include the angles are proportional, then the
triangles are similar.
C
B
G
A
F
E
D
AE
9
BE
3
15
3
бЋЏбЋЏ П­ бЋЏбЋЏ or бЋЏбЋЏ and бЋЏбЋЏ П­ бЋЏбЋЏ or бЋЏбЋЏ
DF
12
CF
4
20
4
AE
DF
BE
CF
By substitution, бЋЏбЋЏ П­ бЋЏбЋЏ. So, by SAS Similarity, б­ќABE Пі б­ќDCF.
www.geometryonline.com/extra_examples/sol
Lesson 6-3 Similar Triangles 299
Like the congruence of triangles, similarity of triangles is reflexive, symmetric,
and transitive.
Theorem 6.3
Similarity of triangles is reflexive, symmetric, and transitive.
Examples:
Reflexive:
б­ќABC Пі б­ќABC
Symmetric:
If б­ќABC Пі б­ќDEF, then б­ќDEF Пі б­ќABC.
Transitive:
If б­ќABC Пі б­ќDEF and б­ќDEF Пі б­ќGHI, then б­ќABC Пі б­ќGHI.
You will prove Theorem 6.3 in Exercise 38.
USE SIMILAR TRIANGLES Similar triangles can be used to solve problems.
Example 2 Parts of Similar Triangles
ALGEBRA Find AE and DE.
BаїЈC
Since а·†
Aа·†
а·†D
а·†, Р„BAE РҐ Р„CDE and
Р„ABE РҐ Р„DCE because they are the
alternate interior angles. By AA Similarity,
б­ќABE Пі б­ќDCE. Using the definition of
AB
DC
C
A xПЄ1
2
E
B
5
xП©5
D
AE
DE
similar polygons, бЋЏбЋЏ П­ бЋЏбЋЏ.
AB
AE
бЋЏ
бЋЏбЋЏ П­ бЋЏбЋЏ
DC
DE
2
xПЄ1
бЋЏбЋЏ П­ бЋЏбЋЏ
5
xП©5
Substitution
2(x П© 5) П­ 5(x ПЄ 1)
Cross products
2x П© 10 П­ 5x ПЄ 5
Distributive Property
ПЄ3x П­ ПЄ15
xП­5
Subtract 5x and 10 from each side.
Divide each side by ПЄ3.
Now find AE and ED.
AE П­ x ПЄ 1
П­ 5 ПЄ 1 or 4
ED П­ x П© 5
П­ 5 П© 5 or 10
Study Tip
Shadow Problems
In shadow problems, we
assume that a right
triangle is formed by the
sun’s ray from the top of
the object to the end of
the shadow.
Object
Shadow
300
Similar triangles can be used to find measurements indirectly.
Example 3 Find a Measurement
INDIRECT MEASUREMENT Nina was
curious about the height of the Eiffel Tower.
She used a 1.2 meter model of the tower
and measured its shadow at 2 P.M. The
length of the shadow was 0.9 meter. Then
she measured the Eiffel Tower’s shadow,
and it was 240 meters. What is the height
of the Eiffel Tower?
Assuming that the sun’s rays form similar
triangles, the following proportion can be
written.
xm
1.2 m
0.9 m
Eiffel Tower shadow length (m)
height of the Eiffel Tower (m)
бЋЏбЋЏбЋЏбЋЏ П­ бЋЏбЋЏбЋЏбЋЏ
height of the model tower (m)
model shadow length (m)
Chapter 6 Proportions and Similarity
240 m
Now substitute the known values and let x be the height of the Eiffel Tower.
x
240
бЋЏбЋЏ П­ бЋЏбЋЏ
1.2
0.9
Substitution
x Рё 0.9 П­ 1.2(240)
0.9x П­ 288
Cross products
Simplify.
x П­ 320
Divide each side by 0.9.
The Eiffel Tower is 320 meters tall.
Concept Check
1. Compare and contrast the tests to prove triangles similar with the tests to prove
triangles congruent.
2. OPEN ENDED Is it possible that б­ќABC is not similar to б­ќRST and that б­ќRST is
not similar to б­ќEFG, but that б­ќABC is similar to б­ќEFG? Explain.
3. FIND THE ERROR Alicia and Jason were writing proportions for the similar
triangles shown at the right.
Alicia
Jason
r
s
бЋЏбЋЏ = бЋЏбЋЏ
k
m
r
m
бЋЏбЋЏ = бЋЏбЋЏ
k
s
rm = ks
rs = km
t
r
k
s
n
m
Who is correct? Explain your reasoning.
Guided Practice
ALGEBRA Identify the similar triangles. Find x and the measures of the
indicated sides.
4. DE
5. AB and DE
C
F
x
E 3 F
B
3
D
A
15
x
E
xПЄ4
45
5
B
A
C
D
Determine whether each pair of triangles is similar. Justify your answer.
E
6.
7.
8.
A
A
25
D
8
D
D
10
21
9
8
F
E 4 F
1
C 5 B
83
E
F
C
7
B
5
8
5
C
A 3 B
Application
9. INDIRECT MEASUREMENT A cell phone tower in a field casts a shadow of
100 feet. At the same time, a 4 foot 6 inch post near the tower casts a shadow of
3 feet 4 inches. Find the height of the tower in feet and inches. (Hint: Make a
drawing.)
Lesson 6-3 Similar Triangles 301
Practice and Apply
For
Exercises
See
Examples
10–17, 26–27
18–21, 28–31
38–41
1
2
3
Extra Practice
Determine whether each pair of triangles is similar. Justify your answer.
10.
11.
N
R
Q 10
R 15 P
7
30
M
45
A
15
12.
E
See page 765.
7
O
75Лљ
7
U
K
R
40Лљ
120Лљ
J
T
15.
V
20Лљ
S
T
J
B
A
9
5
48Лљ
C
W
X
L
S
3
U
120Лљ
J
75Лљ 10
F
14.
K
15
L
17.
E
D
R
A
12.6 m
C
30 m
B
6m
S
42 m
20 m
T
10.5 m
A
V
14
13.
G
16.
14
6
Q 3S
I
8
12
T
21
C
B
ALGEBRA Identify the similar triangles, and find x and the measures of the
indicated sides.
18. AB and BC
19. AB and AC
A
A
xП©3
B
5
E
3
xП©2
D
8E
2x ПЄ 8
B
5
6
C
D
20. BD and EC
F
21. AB and AS
BD ϭ x – 1
CE П­ x + 2
S
x
8
A
C
D
B
7
9
6
A
12
3
B
C
R
C
E
COORDINATE GEOMETRY Triangles ABC and TBS have vertices A(ПЄ2, ПЄ8),
B(4, 4), C(ПЄ2, 7), T(0, ПЄ4), and S(0, 6).
22. Graph the triangles and prove that б­ќABC Пі б­ќTBS.
23. Find the ratio of the perimeters of the two triangles.
Identify each statement as true or false. If false, state why.
24. For every pair of similar triangles, there is only one correspondence of vertices
that will give you correct angle correspondence and segment proportions.
25. If б­ќABC Пі б­ќEFG and б­ќABC Пі б­ќRST, then б­ќEFG Пі б­ќRST.
302 Chapter 6 Proportions and Similarity
Identify the similar triangles in each figure. Explain your answer.
12
A
26. Q
27.
D
6
T
B
R
8
C
E
3
F
S
Use the given information to find each measure.
29. If P
28. If P
а·†R
а·†аїЈ W
а·†X
а·†, WX П­ 10, XY П­ 6,
а·†R
а·†аїЈ K
а·†Lа·†, KN П­ 9, LN П­ 16,
WY П­ 8, RY П­ 5, and PS П­ 3, find
PM П­ 2(KP), find KP, KM, MR, ML,
MN, and PR.
PY, SY, and PQ.
P
Y
L
S
R
W
N
R
K
P
Q
Q
X
IJ
XJ
M
HJ
YJ
30. If бЋЏбЋЏ П­ бЋЏбЋЏ, mР„WXJ П­ 130,
and mР„WZG П­ 20, find mР„YIZ,
mР„JHI, mР„JIH, mР„J, and mР„JHG.
31. If Р„RST is a right angle, S
а·†U
а·†РЊR
а·†T
а·†,
UV
а·†
а·†РЊS
а·†T
а·†, and mР„RTS П­ 47, find
mР„TUV, mР„R, mР„RSU, and mР„SUV.
J
G
H
I
130Лљ
W
S
X
V
20Лљ
Y Z
R
U
T
32. HISTORY The Greek mathematician Thales was the first to measure the height
of a pyramid by using geometry. He showed that the ratio of a pyramid to a
staff was equal to the ratio of one shadow to the other. If a pace is about 3 feet,
approximately how tall was the pyramid at that time?
A
E
Height of
pyramid
Shadow of
pyramid
Shadow
of staff
2 paces
C
B
125 paces
114 paces
33. In the figure at the right, what relationship must
be true of x and y for B
а·†D
а·† and A
а·†E
а·† to be parallel?
Explain.
F
D
3 paces
C
2m
B
2m
A
xm
D
ym
E
Lesson 6-3 Similar Triangles 303
For Exercises 34–38, write the type of proof specified.
PROOF
34. Write a two-column proof to show that if the measures of two sides of a
triangle are proportional to the measures of two corresponding sides of another
triangle and the included angles are congruent, then the triangles are similar.
(Theorem 6.2)
35. a two-column proof
Given: Lа·†P
а·†аїЈ M
а·†N
а·†
LJ
JN
36. a paragraph proof
Given: E
а·†B
а·†РЊ A
а·†C
а·†, B
а·†H
а·†РЊ A
а·†E
а·†,
PJ
JM
Prove: бЋЏбЋЏ П­ бЋЏбЋЏ
L
C
а·†Jа·† РЊ A
а·†E
а·†
Prove: a. б­ќABH Пі б­ќDCB
P
BC
BD
бЋЏбЋЏ П­ бЋЏбЋЏ
BE
BA
b.
J
E
M
J
N
D
H
A
B
C
37. a two-column proof to show that if the measures of the legs of two right
triangles are proportional, the triangles are similar
38. a two-column proof to prove that similarity of triangles is reflexive, symmetric,
and transitive. (Theorem 6.3)
39. SURVEYING Mr. Glover uses a
carpenter’s square, an instrument used
to draw right angles, to find the distance
across a stream. The carpenter’s square
models right angle NOL. He puts the
square on top of a pole that is high
enough to sight along O
а·†L
а·† to point P
across the river. Then he sights along
O
а·†N
а·† to point M. If MK is 1.5 feet and
OK П­ 4.5 feet, find the distance KP
across the stream.
O
L
carpenter’s
square
N
M
K
P
40. The lengths of three sides of triangle ABC are 6 centimeters, 4 centimeters, and
9 centimeters. Triangle DEF is similar to triangle ABC. The length of one of the
sides of triangle DEF is 36 centimeters. What is the greatest perimeter possible
for triangle DEF?
Towers
Completed in 1999, the
Jin Mao Tower is 88
stories tall and sits on a
6-story podium, making it
the tallest building in
China. It is structurally
engineered to tolerate
typhoon winds
and earthquakes.
Source: www.som.com
TOWERS For Exercises 41 and 42, use the following information.
To estimate the height of the Jin Mao
Tower in Shanghai, a tourist sights the
top of the tower in a mirror that is
87.6 meters from the tower. The mirror
is on the ground and faces upward. The
tourist is 0.4 meter from the mirror, and
A
the distance from his eyes to the ground
is about 1.92 meters.
1.92 m
xm
41. How tall is the tower?
42. Why is the mirror reflection a better
way to indirectly measure the tower
than by using shadows?
304 Chapter 6 Proportions and Similarity
Macduff Everton/CORBIS
E
B 0.4 m C
87.6 m
D
43. FORESTRY A hypsometer as shown
can be used to estimate the height of a
tree. Bartolo looks through the straw to
the top of the tree and obtains the
readings given. Find the height of
the tree.
Forester
Foresters plan and manage
the growing, protection,
and harvesting of trees. A
4-year degree in forestry
stressing mathematics,
science, computer science,
communication skills, and
technical forestry is
required for a license in
many states.
Online Research
For information about
a career as a forester,
visit
www.geometryonline.
com/careers
G
Hypsometer
straw D 10 cm
A
AD
F
F
6 cm 1.75 m
E
m
xxm
H
15 m
44. CRITICAL THINKING Suppose you know the height of a flagpole on the beach
of the Chesapeake Bay and that it casts a shadow 4 feet long at 2:00 (EST). You
also know the height of a flagpole on the shoreline of Lake Michigan whose
shadow is hard to measure at 1:00 (CST). Since 2:00 (EST) П­ 1:00 (CST), you
propose the following proportion of heights and lengths to find the length of the
shadow of the Michigan flagpole. Explain whether this proportion will give an
accurate measure.
height of Michigan flagpole
height of Chesapeake flagpole
бЋЏбЋЏбЋЏбЋЏ П­ бЋЏбЋЏбЋЏбЋЏ
shadow of Chesapeake flagpole
shadow of Michigan flagpole
COORDINATE GEOMETRY For Exercises 45 and 46, use the following
information.
The coordinates of б­ќABC are A(ПЄ10, 6), B(ПЄ2, 4), and C(ПЄ4, ПЄ2). Point D(6, 2)
lies on бџ­бџ®бџ¬
AB .
D.
45. Graph б­ќABC, point D, and draw B
а·†а·†
46. Where should a point E be located so that б­ќABC Пі б­ќADE?
47. CRITICAL THINKING The altitude C
а·†D
а·† from
the right angle C in triangle ABC forms two
triangles. Triangle ABC is similar to the two
triangles formed, and the two triangles formed
are similar to each other. Write three similarity
statements about these triangles. Why are the
triangles similar to each other?
C
B
D
A
48. WRITING IN MATH
Answer the question that was posed at the beginning of
the lesson.
How do engineers use geometry?
SOL/EOC Practice
Standardized
Test Practice
Include the following in your answer:
• why engineers use triangles in construction, and
• why you think the pressure applied to the ground from the Eiffel Tower was
so small.
49. If E
а·†B
а·†аїЈ D
а·†C
а·†, find x.
A
C
A
9.5
B
4
D
5
2
xПЄ2
10
E
4
xП©3
6
x
xПЄ2
D
50. ALGEBRA Solve бЋЏбЋЏ П­ бЋЏбЋЏ.
A
6 or 1
B
6 or ПЄ1
C
3 or 2
D
ПЄ3 or 2
www.geometryonline.com/self_check_quiz /sol
B
5
C
Lesson 6-3 Similar Triangles 305
Lawrence Migdale/Stock Boston
Maintain Your Skills
Mixed Review
Each pair of polygons is similar. Write a similarity statement, find x, the measures
of the indicated sides, and the scale factor. (Lesson 6-2)
51. B
52. E
а·†C
а·†, P
а·†Sа·†
а·†Fа·†, X
а·†Z
а·†
F
C
1.4
P
D
6x
Q
x
X
22.5
0.7
E
Y
2.2
S 0.7
3.2
B
R
Z
10
25
G
7.5
A
Solve each proportion. (Lesson 6-1)
3
1
53. бЋЏбЋЏ П­ бЋЏбЋЏ
y
m
20
55. бЋЏбЋЏ П­ бЋЏбЋЏ
6
7
54. бЋЏбЋЏ П­ бЋЏбЋЏ
15
8
b
28
21
16
9
56. бЋЏбЋЏ П­ бЋЏбЋЏ
7
s
57. COORDINATE GEOMETRY б­ќABC has vertices A(ПЄ3, ПЄ9), B(5, 11), and
C(9, ПЄ1). A
а·†T
а·† is a median from A to B
а·†C
а·†. Determine whether A
а·†T
а·† is an altitude.
(Lesson 5-1)
58. ROLLER COASTERS The sign in front of the Electric Storm roller coaster states
ALL riders must be at least 54 inches tall to ride. If Adam is 5 feet 8 inches tall, can
he ride the Electric Storm? Which law of logic leads you to this conclusion?
(Lesson 2-4)
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Find the coordinates of the midpoint of the segment whose
endpoints are given. (To review finding coordinates of midpoints, see Lesson 1-3.)
59. (2, 15), (9, 11)
60. (ПЄ4, 4), (2, ПЄ12)
61. (0, 8), (7, ПЄ13)
P ractice Quiz 1
Lessons 6-1 through 6-3
Determine whether each pair of figures is similar. Justify your answer. (Lesson 6-2)
B
1.
2. A
N
L
5.5
A
C
2
4
F
6
6.5
1
D
3
C
E
5
B
M
Identify the similar triangles. Find x and the measures of the indicated sides. (Lesson 6-3)
B
3. A
E, а·†
Dа·†
E
4. а·†
PT
ST
а·†а·†
а·†, а·†
а·†
P
A
6
T
E
15
25
x
10
3x ПЄ 2
12
D
C
R 10 S 5 Q
5. MAPS The scale on a map shows that 1.5 centimeters represents 100 miles. If the
distance on the map from Atlanta, Georgia, to Los Angeles, California, is 29.2 centimeters,
approximately how many miles apart are the two cities? (Lesson 6-1)
306 Chapter 6 Proportions and Similarity
Parallel Lines and
Proportional Parts
• Use proportional parts of triangles.
• Divide a segment into parts.
• midsegment
Lake Michigan
do city planners use geometry?
Vocabulary
Street maps frequently have parallel and
perpendicular lines. In Chicago, because of
Lake Michigan, Lake Shore Drive runs at an
angle between Oak Street and Ontario Street.
City planners need to take this angle into
account when determining dimensions of
available land along Lake Shore Drive.
ore
e
Driv
Ontario St.
Michigan Avenue
Delaware Pl.
e Sh
Lak
Oak St.
Walton St.
PROPORTIONAL PARTS OF TRIANGLES Nonparallel transversals that
intersect parallel lines can be extended to form similar triangles. So the sides of the
triangles are proportional.
Theorem 6.4
Triangle Proportionality Theorem If a line is parallel to
one side of a triangle and intersects the other two sides
in two distinct points, then it separates these sides into
segments of proportional lengths.
Example: If а·†
BD
а·†аїЈ
Proof
A
Theorem 6.4
Given: а·†
BD
а·† аїЈ а·†A
а·†E
а·†
Overlapping
Triangles
DE
BA
Prove: бЋЏбЋЏ П­ бЋЏбЋЏ
CB
B
DE
BA
A
E, бЋЏбЋЏ П­ бЋЏбЋЏ.
а·†а·†
CD
CB
Study Tip
Trace two copies of б­ќACE.
Cut along а·†BD
а·† to form
б­ќBCD. Now б­ќACE and
б­ќBCD are no longer
overlapping. Place the
triangles side-by-side to
compare corresponding
angles and sides.
C
D
E
C
B 1
CD
4
2 D
3
A
E
Paragraph Proof:
Since а·†B
а·†D
а·† аїЈ а·†A
а·†E
а·†, Р„4 РҐ Р„1 and Р„3 РҐ Р„2 because they are corresponding angles.
Then, by AA Similarity, б­ќACE Пі б­ќBCD. From the definition of similar polygons,
CE
CA
бЋЏбЋЏ П­ бЋЏбЋЏ. By the Segment Addition Postulate, CA П­ BA П© CB and CE П­ DE П© CD.
CD
CB
Substituting for CA and CE in the ratio, we get the following proportion.
BA П© CB
DE П© CD
бЋЏбЋЏ П­ бЋЏбЋЏ
CB
CD
DE
BA
CB
CD
бЋЏбЋЏ П© бЋЏбЋЏ П­ бЋЏбЋЏ П© бЋЏбЋЏ
CD
CB
CB
CD
DE
BA
бЋЏбЋЏ П© 1 П­ бЋЏбЋЏ П© 1
CD
CB
DE
BA
бЋЏбЋЏ П­ бЋЏбЋЏ
CD
CB
Rewrite as a sum.
CB
CD
бЋЏбЋЏ П­ 1 and бЋЏбЋЏ П­ 1
CB
CD
Subtract 1 from each side.
Lesson 6-4 Parallel Lines and Proportional Parts 307
Example 1 Find the Length of a Side
In б­ќEFG, а·†
HL
EF
а·†аїЈа·†
а·†, EH П­ 9, HG П­ 21, and FL П­ 6. Find LG.
Study Tip
From the Triangle Proportionality Theorem, бЋЏбЋЏ П­ бЋЏбЋЏ.
Using Fractions
Substitute the known measures.
You can also rewrite
9
3
бЋЏбЋЏ as бЋЏбЋЏ. Then use your
21
7
knowledge of fractions
to find the missing
denominator.
П«2
3
6
бЋЏбЋЏ П­ бЋЏбЋЏ
7
?
П«2
The correct denominator
is 14.
EH
HG
FL
LG
E
9
6
бЋЏбЋЏ П­ бЋЏбЋЏ
21
LG
F
L
H
9(LG) П­ (21)6 Cross products
9(LG) П­ 126
Multiply.
LG П­ 14
G
Divide each side by 9.
Proportional parts of a triangle can also be used to prove the converse of Theorem 6.4.
Theorem 6.5
Converse of the Triangle Proportionality Theorem
C
If a line intersects two sides of a triangle and
separates the sides into corresponding
segments of proportional lengths, then
the line is parallel to the third side.
E
DE
BA
BD
Example: If бЋЏбЋЏ П­ бЋЏбЋЏ, then а·†
а·†аїЈA
а·†E
а·†.
CB
D
B
A
CD
You will prove Theorem 6.5 in Exercise 38.
Example 2 Determine Parallel Lines
In б­ќHKM, HM П­ 15, HN П­ 10, and а·†
HJа·† is twice the
NJа·† аїЈ а·†
length of Jа·†K
MK
а·†. Determine whether а·†
а·†. Explain.
HM П­ HN П© NM Segment Addition Postulate
15 П­ 10 П© NM
5 П­ NM
H
N
J
M
HM П­ 15, HN П­ 10
K
Subtract 10 from each side.
HN
NM
HJ
JK
In order to show а·†
Nа·†J аїЈ а·†
MK
а·†, we must show that бЋЏбЋЏ П­ бЋЏбЋЏ. HN П­ 10 and NM П­
HN
NM
10
5
HJ
JK
2x
x
HM ПЄ HN or 5. So бЋЏбЋЏ П­ бЋЏбЋЏ or 2. Let JK П­ x. Then HJ П­ 2x. So, бЋЏбЋЏ П­ бЋЏбЋЏ or 2.
HN
NM
HJ
JK
NJа·† аїЈ а·†
Thus, бЋЏбЋЏ П­ бЋЏбЋЏ П­ 2. Since the sides have proportional lengths, а·†
MK
а·†.
A midsegment of a triangle is a segment whose endpoints are the midpoints of
two sides of the triangle.
Theorem 6.6
Triangle Midsegment Theorem A midsegment of a
C
triangle is parallel to one side of the triangle, and its
length is one-half the length of that side.
C and E
Example: If B and D are midpoints of а·†
Aа·†
а·†C
а·†
1
respectively, B
DаїЈA
а·†а·†
а·†E
а·† and BD П­ бЋЏбЋЏ AE.
2
You will prove Theorem 6.6 in Exercise 39.
308
Chapter 6 Proportions and Similarity
B
A
D
E
Example 3 Midsegment of a Triangle
Study Tip
Look Back
To review the Distance
and Midpoint Formulas,
see Lesson 1-3.
Triangle ABC has vertices A(ПЄ4, 1), B(8, ПЄ1),
and C(ПЄ2, 9). а·†
DE
а·† is a midsegment of б­ќABC.
a. Find the coordinates of D and E.
Use the Midpoint Formula to find the
midpoints of A
а·†B
а·† and C
а·†B
а·†.
C (–2, 9) y
E
ПЄ4 П© 8 1 П© (ПЄ1)
2
2
DО‚бЋЏбЋЏ, бЋЏбЋЏОѓ П­ D(2, 0)
A (–4, 1)
ПЄ2 П© 8 9 П© (ПЄ1)
2
2
EО‚бЋЏбЋЏ, бЋЏбЋЏОѓ П­ E(3, 4)
O
x
D
B (8, –1)
b. Verify that A
а·†C
а·† is parallel to D
а·†E
а·†.
If the slopes of A
DE
а·†C
а·† and D
а·†E
а·† are equal, A
а·†C
а·†аїЈа·†
а·†.
9ПЄ1
ПЄ2 ПЄ (ПЄ4)
4ПЄ0
slope of а·†
DE
а·† П­ бЋЏбЋЏ or 4
3ПЄ2
slope of A
а·†C
а·† П­ бЋЏбЋЏ or 4
DE
Because the slopes of A
а·†C
а·† and D
а·†E
а·† are equal, A
а·†C
а·†аїЈа·†
а·†.
1
c. Verify that DE П­ бЋЏбЋЏAC.
2
First, use the Distance Formula to find AC and DE.
AC П­ Н™[ПЄ2
ПЄа·†
(ПЄ4)]2а·†
П© (9а·†
ПЄ 1)2
а·†
DE П­ Н™а·†
(3 ПЄ2 а·†
)2 П© (4а·†
ПЄ 0)2
П­ Н™а·†
4 П© 64
П­ Н™а·†
1 П© 16
П­ Н™68
а·†
П­ Н™17
а·†
а·†
Н™17
DE
бЋЏбЋЏ П­ бЋЏ
AC
а·†
Н™68
П­
DE
AC
Ίᎏ๶14ᎏ or ᎏ12ᎏ
1
2
1
2
If бЋЏбЋЏ П­ бЋЏбЋЏ, then DE П­ бЋЏбЋЏAC.
DIVIDE SEGMENTS PROPORTIONALLY We have seen that parallel lines
cut the sides of a triangle into proportional parts. Three or more parallel lines also
separate transversals into proportional parts. If the ratio is 1, they separate the
transversals into congruent parts.
Study Tip
Three Parallel
Lines
Corollary 6.1 is a special
case of Theorem 6.4. In
some drawings, the
transversals are not
shown to intersect. But,
if extended, they will
intersect and therefore,
form triangles with each
parallel line and the
transversals.
Corollaries
6.1
If three or more parallel lines intersect
two transversals, then they cut off the
transversals proportionally.
D
AB
DE
бџ­бџ®бџ¬ аїЈ бџ­бџ®бџ¬
Example: If DA
EB аїЈ бџ­бџ®бџ¬
FC , then бЋЏбЋЏ П­ бЋЏбЋЏ,
A
BC
AC
BC
AC
DF
бЋЏ П­ бЋЏбЋЏ, and бЋЏбЋЏ П­ бЋЏбЋЏ.
DF
EF
BC
EF
6.2
EF
E
B
F
C
If three or more parallel lines cut off congruent segments on one
transversal, then they cut off congruent segments on every transversal.
Example: If а·†
AB
BC
DE
EFа·†.
а·†РҐа·†
а·†, then а·†
а·†РҐа·†
www.geometryonline.com/extra_examples /sol
Lesson 6-4 Parallel Lines and Proportional Parts 309
Example 4 Proportional Segments
Maps
Source: www.nima.mil
Delaware to Walton
бЋЏбЋЏбЋЏ
Oak to Ontario
x
бЋЏбЋЏ
4430
3800 Рё x П­ 411(4430)
3800x П­ 1,820,730
x П­ 479
rive
Delaware to Walton
бЋЏбЋЏбЋЏ П­
Oak to Ontario
411
бЋЏбЋЏ П­
3800
eD
hor
Lake Shore Drive
Delaware
Pl.
eS
Lak
Michigan Ave.
Michigan Avenue
Modern map-making
techniques use images
taken from space to
produce accurate
representations on paper.
In February 2000, the
crew of the space shuttle
Endeavor collected a
trillion radar images of
46 million square miles
of Earth.
MAPS Refer to the map at the beginning of the lesson. The streets from Oak
Street to Ontario Street are all parallel to each other. The distance from Oak
Street to Ontario along Michigan Avenue is about 3800 feet. The distance
between the same two streets along Lake Shore Drive is about 4430 feet. If
the distance from Delaware Place to Walton Street along Michigan Avenue
is about 411 feet, what is the distance between those streets along Lake
Shore Drive?
Make a sketch of the streets in
the problem. Notice that the streets
Lake Michigan
form the bottom portion of a triangle
Walton Oak
that is cut by parallel lines. So you
St.
St.
can use the Triangle Proportionality
Theorem.
Ontario St.
Triangle Proportionality Theorem
Substitution
Cross products
Multiply.
Divide each side by 3800.
The distance from Delaware Place to Oak Street along Lake Shore Drive is
about 479 feet.
Study Tip
Locus
The locus of points in a
plane equidistant from
two parallel lines is a line
that lies between the lines
and is parallel to them. In
бџ­бџ®бџ¬ is the locus
Example 5, BE
of points in the plane
бџ­бџ®бџ¬ and
equidistant from AD
бџ­бџ®бџ¬.
CF
Example 5 Congruent Segments
Find x and y.
To find x:
AB П­ BC
A
Given
3x ПЄ 4 П­ 6 ПЄ 2x Substitution
5x ПЄ 4 П­ 6
5x П­ 10
xП­2
3x ПЄ 4
6 ПЄ 2x
D
B
E
C
F
5
3yП©1
3y
Add 2x to each side.
Add 4 to each side.
Divide each side by 5.
To find y:
D
EРҐа·†
EF
а·†а·†
а·†
Parallel lines that cut off congruent segments on one transversal cut off
congruent segments on every transversal.
DE РҐ EF
Definition of congruent segments
5
3y П­ бЋЏбЋЏy П© 1
3
9y П­ 5y П© 3
Multiply each side by 3 to eliminate the denominator.
4y П­ 3
Subtract 5y from each side.
3
y П­ бЋЏбЋЏ
4
310
Chapter 6 Proportions and Similarity
JPL/NIMA/NASA
Substitution
Divide each side by 4.
It is possible to separate a segment into two congruent parts by constructing the
perpendicular bisector of a segment. However, a segment cannot be separated into
three congruent parts by constructing perpendicular bisectors. To do this, you must
use parallel lines and the similarity theorems from this lesson.
Trisect a Segment
а·†B
а·† to be trisected.
1 Draw A
бџ®бџ¬.
Then draw AM
A
B
а·†B
а·†. Then construct lines
3 Draw Z
2 With the compass at A, mark
бџ®бџ¬ at
off an arc that intersects AM
X. Use the same compass
setting to construct а·†
XY
а·† and Y
а·†Z
а·†
congruent to а·†
AX
а·†.
A
through Y and X that are
parallel to а·†
ZB
а·†. Label the
intersection points on A
а·†B
а·†
as P and Q.
P
A
B
Q
B
X
X
Y
Y
Z
M
Z
M
M
Conclusion: Because parallel lines cut off congruent segments on transversals, A
PРҐа·†
PQ
QB
а·†а·†
а·†РҐа·†
а·†.
Concept Check
Guided Practice
1. Explain how you would know if a line that intersects two sides of a triangle is
parallel to the third side.
2. OPEN ENDED Draw two segments that are intersected by three lines so that the
parts are proportional. Then draw a counterexample.
3. Compare and contrast Corollary 6.1 and Corollary 6.2.
For Exercises 4 and 5, refer to б­ќRST.
4. If RL П­ 5, RT П­ 9, and WS П­ 6, find RW.
5. If TR П­ 8, LR П­ 3, and RW П­ 6, find WS.
T
L
W
R
COORDINATE GEOMETRY For Exercises
6–8, use the following information.
Triangle ABC has vertices A(ПЄ2, 6), B(ПЄ4, 0),
and C(10, 0). а·†
DE
а·† is a midsegment.
y
A(ПЄ2, 6)
D
E
6. Find the coordinates of D and E.
7. Verify that D
а·†E
а·† is parallel to B
а·†C
а·†.
B(ПЄ4, 0)
S
O
C(10, 0) x
1
8. Verify that DE П­ бЋЏбЋЏBC.
2
M
9. In б­ќMQP, MP П­ 25, MN П­ 9,
MR П­ 4.5, and MQ П­ 12.5.
QP
Determine whether а·†
RN
а·†аїЈа·†
а·†.
Justify your answer.
R
Q
N
P
Lesson 6-4 Parallel Lines and Proportional Parts 311
10. In б­ќACE, ED П­ 8, DC П­ 20, BC П­ 25,
and AB П­ 12. Determine whether а·†
DB
AE
а·†аїЈа·†
а·†.
E
C
B
A
11. Find x and y.
D
12. Find x and y.
5y
20 ПЄ 5x
3yП©2
5
2
3xПЄ4
13. MAPS The distance along Talbot Road
from the Triangle Park entrance to the
Walkthrough is 880 yards. The distance
along Talbot Road from the Walkthrough to
Clay Road is 1408 yards. The distance along
Woodbury Avenue from the Walkthrough to
Clay Road is 1760 yards. If the Walkthrough
is parallel to Clay Road, find the distance
from the entrance to the Walkthrough along
Woodbury.
Triangle Park
Entrance
od
Wo
ry
bu
Talbot Rd.
Application
1
3xП©2
y
2x П© 6
7
3y П© 8
e.
Av
Walkthrough
Clay Rd.
Practice and Apply
For
Exercises
See
Examples
14–19
20–26
27, 28
35–37, 43
33, 34
1
2
3
4
5
Extra Practice
For Exercises 14 and 15, refer to б­ќXYZ.
14. If XM П­ 4, XN П­ 6, and NZ П­ 9, find XY.
15. If XN П­ t ПЄ 2, NZ П­ t П© 1, XM П­ 2, and
XY П­ 10, solve for t.
16. If DB П­ 24, AE П­ 3, and EC П­ 18,
find AD.
A
See page 765.
D
Y
M
X
N
Z
17. Find x and ED if AE П­ 3, AB П­ 2,
BC П­ 6, and ED П­ 2x ПЄ 3.
D
E
E
A
B
B
C
C
18. Find x, AC, and CD if AC П­ x ПЄ 3,
BE П­ 20, AB П­ 16, and CD П­ x П© 5.
D
19. Find BC, FE, CD, and DE if
AB П­ 6, AF П­ 8, BC П­ x, CD П­ y,
10
DE П­ 2y ПЄ 3, and FE П­ x П© бЋЏбЋЏ.
3
A
C
A
B
F
B
E
C
312 Chapter 6 Proportions and Similarity
D
E
Find x so that а·†
GJа·† аїЈ а·†
FK
а·†.
20. GF П­ 12, HG П­ 6, HJ П­ 8, JK П­ x ПЄ 4
H
21. HJ П­ x ПЄ 5, JK П­ 15, FG П­ 18, HG П­ x ПЄ 4
G
J
22. GH ϭ x ϩ 3.5, HJ ϭ x – 8.5, FH ϭ 21, HK ϭ 7
F
K
Determine whether а·†
QT
RS
а·†аїЈа·†
а·†. Justify your answer.
23. PR П­ 30, PQ П­ 9, PT П­ 12, and PS П­ 18
R
Q
24. QR П­ 22, RP П­ 65, and SP is 3 times TS.
25. TS П­ 8.6, PS П­ 12.9, and PQ is half RQ.
P
T
S
26. PQ П­ 34.88, RQ П­ 18.32, PS П­ 33.25, and TS П­ 11.45
27. Find the length of B
DE
а·†C
а·† if B
а·†C
а·†аїЈа·†
а·†
and а·†
DE
а·† is a midsegment of б­ќABC.
28. Show that W
TS
а·†M
а·†аїЈа·†
а·† and determine
whether а·†
WM
а·† is a midsegment.
y
y
A (ПЄ3, 4)
E ( 4, 3)
C
22
D ( 1, 1)
O
T ( 11, 26)
26
18
x
( 3, 14)
14
B
S ( 17, 20)
W
M ( 5, 12)
10
(
)
6 R ПЄ1, 8
2
O
ПЄ2
2
COORDINATE GEOMETRY For Exercises 29
and 30, use the following information.
Triangle ABC has vertices A(ПЄ1, 6), B(ПЄ4, ПЄ3),
and C(7, ПЄ5). D
а·†E
а·† is a midsegment.
6
10 14 18
x
y
A (ПЄ1, 6)
4
D
29. Verify that D
а·†E
а·† is parallel to A
а·†B
а·†.
8 x
O
1
30. Verify that DE П­ бЋЏбЋЏAB.
2
B (ПЄ4, ПЄ3)
E
31. COORDINATE GEOMETRY Given A(2, 12)
and B(5, 0), find the coordinates of P such
that P separates A
а·†B
а·† into two parts with a
ratio of 2 to 1.
C (7, ПЄ5)
32. COORDINATE GEOMETRY In б­ќLMN, P
а·†R
а·† divides N
а·†L
а·† and M
а·†N
а·† proportionally.
LP
2
If the vertices are N(8, 20), P(11, 16), and R(3, 8) and бЋЏбЋЏ П­ бЋЏбЋЏ, find the
PN
1
coordinates of L and M.
ALGEBRA
33. 5
Find x and y.
34.
3 x П© 11
3y ПЄ 9
2x П© 3
xП©2
2y П© 6
6ПЄx
4
3y П© 1
2y
Lesson 6-4 Parallel Lines and Proportional Parts 313
CONSTRUCTION For Exercises 35–37, use
the following information and drawing.
Two poles, 30 feet and 50 feet tall, are 40 feet
apart and perpendicular to the ground. The
poles are supported by wires attached from
the top of each pole to the bottom of the
other, as in the figure. A coupling is placed
at C where the two wires cross.
50 ft
C
30 ft
a ft
35. Find x, the distance from C to the
taller pole.
x ft
40 ft
36. How high above the ground is the coupling?
37. How far down the wire from the smaller pole is the coupling?
PROOF
Write a two-column proof of each theorem.
39. Theorem 6.6
38. Theorem 6.5
CONSTRUCTION
Construct each segment as directed.
40. a segment 8 centimeters long, separated into three congruent segments
41. a segment separated into four congruent segments
42. a segment separated into two segments in which their lengths have a ratio
of 1 to 4
43. REAL ESTATE In Lake Creek, the lots
on which houses are to be built are laid
out as shown. What is the lake frontage
for each of the five lots if the total frontage
is 135.6 meters?
Lake Creek Drive
44. CRITICAL THINKING Copy the figure
that accompanies Corollary 6.1 on page
309. Draw D
а·†C
а·†. Let G be the intersection
point of D
а·†C
а·† and B
а·†E
а·†. Using that segment,
AB
BC
20 m
22 m
25 m
18 m
28 m
um
wm
xm
ym
zm
DE
EF
explain how you could prove бЋЏбЋЏ П­ бЋЏбЋЏ.
45. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson.
How do city planners use geometry?
Include the following in your answer:
• why maps are important to city planners, and
• what geometry facts a city planner needs to know to explain why the block
between Chestnut and Pearson is longer on Lake Shore Drive than on
Michigan Avenue.
SOL/EOC Practice
Standardized
Test Practice
12 cm
46. Find x.
A 16
C 24
B
D
16.8
28.4
x cm
18 cm
42 cm
47. GRID IN The average of a and b is 18, and the ratio of a to b is 5 to 4. What is
the value of a ПЄ b?
314 Chapter 6 Proportions and Similarity
Extending
the Lesson
48. MIDPOINTS IN POLYGONS Draw any quadrilateral ABCD on a coordinate
A,
plane. Points E, F, G, and H are midpoints of A
а·†B
а·†, B
а·†C
а·†, C
а·†D
а·†, and D
а·†а·†
respectively.
a. Connect the midpoints to form quadrilateral EFGH. Describe what you know
about the sides of quadrilateral EFGH.
b. Will the same reasoning work with five-sided polygons? Explain why or
why not.
Maintain Your Skills
Mixed Review
Determine whether each pair of triangles is similar. Justify your answer. (Lesson 6-3)
49.
50.
51.
9
6
8
12
38Лљ
16
72Лљ 66Лљ
12
Each pair of polygons is similar. Find x and y. (Lesson 6-2)
52.
53.
66Лљ
9
14
x
9
20
y
7
x
18
y
14
21
Determine the relationship between the measures of the given angles. (Lesson 5-2)
B
54. Р„ADB, Р„ABD
15
55. Р„ABD, Р„BAD
A
13
9
56. Р„BCD, Р„CDB
57. Р„CBD, Р„BCD
12
D
10
C
ARCHITECTURE For Exercises 58 and 59, use the following information.
The geodesic dome was developed by Buckminster Fuller in
the 1940s as an energy-efficient building. The figure at the right
shows the basic structure of one geodesic dome. (Lesson 4-1)
58. How many equilateral triangles are in the figure?
59. How many obtuse triangles are in the figure?
Determine the truth value of the following statement for each set of conditions.
If you have a fever, then you are sick. (Lesson 2-3)
60. You do not have a fever, and you are sick.
61. You have a fever, and you are not sick.
62. You do not have a fever, and you are not sick.
63. You have a fever, and you are sick.
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Write all the pairs of corresponding parts for each pair of
congruent triangles. (To review corresponding congruent parts, see Lesson 4-3.)
64. б­ќABC РҐ б­ќDEF
65. б­ќRST РҐ б­ќXYZ
66. б­ќPQR РҐ б­ќKLM
www.geometryonline.com/self_check_quiz/sol
Lesson 6-4 Parallel Lines and Proportional Parts 315
Parts of Similar Triangles
Virginia SOL Standard G.5a The student will investigate and identify … similarity relationships between triangles
Standard G.14a The student will use proportional reasoning to solve practical problems, given similar
geometric objects;
• Recognize and use proportional relationships of
corresponding perimeters of similar triangles.
• Recognize and use proportional relationships of
corresponding angle bisectors, altitudes, and
medians of similar triangles.
is geometry related to photography?
The camera lens was 6.16 meters from this Dale Chihuly
glass sculpture when this photograph was taken. The
image on the film is 35 millimeters tall. Similar triangles
enable us to find the height of the actual sculpture.
PERIMETERS Triangle ABC is similar to б­ќDEF
with a scale factor of 1 : 3. You can use variables and
the scale factor to compare their perimeters. Let
the measures of the sides of б­ќABC be a, b, and c.
The measures of the corresponding sides of
б­ќDEF would be 3a, 3b, and 3c.
perimeter of б­ќABC
бЋЏбЋЏбЋЏ
perimeter of б­ќDEF
Dale Chihuly
Dale Chihuly (1941– ), born
in Tacoma, Washington, is
widely recognized as one
of the greatest glass artists
in the world. His sculptures
are made of hundreds of
pieces of hand-blown glass
that are assembled to
resemble patterns in
nature.
D
A
3b
b
C
3c
c
a
B
F
E
3a
aП©bП©c
3a П© 3b П© 3c
1(a П© b П© c)
1
П­ бЋЏбЋЏ or бЋЏбЋЏ
3(a П© b П© c)
3
П­ бЋЏбЋЏ
The perimeters are in the same proportion as the side measures of the two similar
figures. This suggests Theorem 6.7, the Proportional Perimeters Theorem.
Theorem 6.7
Proportional Perimeters Theorem If two triangles are similar, then the perimeters
are proportional to the measures of corresponding sides.
You will prove Theorem 6.7 in Exercise 8.
Example 1 Perimeters of Similar Triangles
If б­ќLMN Пі б­ќQRS, QR П­ 35, RS П­ 37, SQ П­ 12, and
NL П­ 5, find the perimeter of б­ќLMN.
Let x represent the perimeter of б­ќLMN. The
perimeter of б­ќQRS П­ 35 П© 37 П© 12 or 84.
perimeter of б­ќLMN
NL
бЋЏбЋЏ П­ бЋЏбЋЏбЋЏ
perimeter of б­ќQRS
SQ
5
x
бЋЏбЋЏ П­ бЋЏбЋЏ
12
84
12x П­ 420
x П­ 35
(l)Kelly-Mooney Photography/CORBIS, (r)Pierre Burnaugh/PhotoEdit
5
Proportional Perimeter Theorem
Substitution
Cross products
Divide each side by 12.
M
L
S
12
The perimeter of б­ќLMN is 35 units.
316 Chapter 6 Proportions and Similarity
N
Q
37
35
R
SPECIAL SEGMENTS OF SIMILAR TRIANGLES Think about a triangle
drawn on a piece of paper being placed on a copy machine and either enlarged or
reduced. The copy is similar to the original triangle. Now suppose you drew in
special segments of a triangle, such as the altitudes, medians, or angle bisectors, on
the original. When you enlarge or reduce that original triangle, all of those segments
are enlarged or reduced at the same rate. This conjecture is formally stated in
Theorems 6.8, 6.9, and 6.10.
Theorems
6.8
Special Segments of Similar Triangles
If two triangles are similar, then the
measures of the corresponding altitudes
are proportional to the measures of the
corresponding sides.
U
Q
T
Abbreviation: Пі б­ќs have corr. altitudes
P
proportional to the corr. sides.
6.9
R
QA
PR
QR
PQ
бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ
UW
TV
UV
TU
If two triangles are similar, then the
measures of the corresponding angle
bisectors of the triangles are proportional
to the measures of the corresponding sides.
Abbreviation: Пі б­ќs have corr. Р„ bisectors
U
Q
T
P
proportional to the corr. sides.
6.10
A
W V
B
X
V
R
QB
PR
QR
PQ
бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ
UX
TV
UV
TU
If two triangles are similar, then the
measures of the corresponding medians
are proportional to the measures of the
corresponding sides.
Abbreviation: Пі б­ќs have corr. medians
U
Q
T
P
M
Y
V
R
proportional to the corr. sides.
PR
QR
QM
PQ
бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ
TV
UV
UY
TU
You will prove Theorems 6.8 and 6.10 in Exercises 30 and 31, respectively.
Example 2 Write a Proof
Write a paragraph proof of Theorem 6.9.
Since the corresponding angles to be bisected are chosen at random, we need not
prove this for every pair of bisectors.
Given: б­ќRTS Пі б­ќEGF
TA
а·†
а·† and G
а·†B
а·† are angle bisectors.
G
T
TA
RT
Prove: бЋЏбЋЏ П­ бЋЏбЋЏ
GB
E
EG
B
F
Paragraph Proof: Because corresponding
R
S
A
angles of similar triangles are congruent,
Р„R РҐ Р„E and Р„RTS РҐ Р„EGF. Since Р„RTS
1
1
and Р„EGF are bisected, we know that бЋЏбЋЏmР„RTS П­ бЋЏбЋЏmР„EGF or mР„RTA П­ mР„EGB.
2
2
TA
GB
RT
EG
This makes Р„RTF РҐ Р„EGB and б­ќRTF Пі б­ќEGB by AA Similarity. Thus, бЋЏбЋЏ П­ бЋЏбЋЏ.
www.geometryonline.com/extra_examples/sol
Lesson 6-5 Parts of Similar Triangles 317
Example 3 Medians of Similar Triangles
In the figure, б­ќABC Пі б­ќDEF. а·†
BG
а·† is a median of
б­ќABC, and а·†
EH
а·† is a median of б­ќDEF. Find EH
if BC П­ 30, BG П­ 15, and EF П­ 15.
Let x represent EH.
BG
BC
бЋЏбЋЏ П­ бЋЏбЋЏ
EH
EF
15
30
бЋЏбЋЏ П­ бЋЏбЋЏ
x
15
30x П­ 225
x П­ 7.5
B
A
C
G
E
Write a proportion.
D
BG П­ 15, EH П­ x, BC П­ 30, and EF П­ 15
H
F
Cross products
Divide each side by 30.
Thus, EH П­ 7.5.
The theorems about the relationships of special segments in similar triangles can
be used to solve real-life problems.
Photography
Example 4 Solve Problems with Similar Triangles
PHOTOGRAPHY Refer to the application at the beginning of the lesson. The
drawing below illustrates the position of the camera and the distance from the
lens of the camera to the film. Find the height of the sculpture.
A
film
F
35 mm
The first consumeroriented digital cameras
were produced for sale in
1994 with a 640 П« 480
pixel resolution. In 2002, a
3.3-megapixel camera
could take a picture with
2048 П« 1536 pixel
resolution, which is a
sharper picture than most
computer monitors can
display.
Source: www.howstuffworks.com
Lens
H
C
6.16 m
G
X
E
42 mm
B
Not drawn
to scale
б­ќABC and б­ќEFC are similar. The distance from the lens to the film in the camera
is CH П­ 42 mm. а·†
CG
а·† and C
а·†H
а·† are altitudes of б­ќABC and б­ќEFC, respectively. If
two triangles are similar, then the measures of the corresponding altitudes are
proportional to the measures of the corresponding sides. This leads to the
GC
AB
proportion бЋЏбЋЏ П­ бЋЏбЋЏ.
EF
GC
AB
бЋЏбЋЏ П­ бЋЏбЋЏ
HC
EF
xm
6.16 m
бЋЏбЋЏ П­ бЋЏбЋЏ
35 mm
42 mm
x Рё 42 П­ 35(6.16)
42x П­ 215.6
x П· 5.13
HC
Write the proportion.
AB П­ x m, EF П­ 35 m, GC П­ 6.16 m, HC П­ 42 mm
Cross products
Simplify.
Divide each side by 42.
The sculpture is about 5.13 meters tall.
318
Chapter 6 Proportions and Similarity
Beth A. Keiser/AP/Wide World Photos
An angle bisector also divides the side of the triangle opposite the angle
proportionally.
Theorem 6.11
C
Angle Bisector Theorem An angle bisector in a triangle separates
the opposite side into segments that have the same ratio as the
other two sides.
в†ђ segments with vertex A
в†ђ segments with vertex B
AD
AC
Example: бЋЏбЋЏ П­ бЋЏбЋЏ
DB
BC
A
D
B
You will prove this theorem in Exercise 32.
Concept Check
1. Explain what must be true about б­ќABC and б­ќMNQ
AD
BA
before you can conclude that бЋЏбЋЏ П­ бЋЏбЋЏ.
MR
A
M
NM
B
N
C
D
R
Q
2. OPEN ENDED The perimeter of one triangle is 24 centimeters, and the perimeter
of a second triangle is 36 centimeters. If the length of one side of the smaller
triangle is 6, find possible lengths of the other sides of the triangles so that they
are similar.
Guided Practice
Find the perimeter of the given triangle.
3. б­ќDEF, if б­ќABC Пі б­ќDEF, AB П­ 5,
4. б­ќWZX, if б­ќWZX Пі б­ќSRT, ST П­ 6,
WX П­ 5, and the perimeter of
BC П­ 6, AC П­ 7, and DE П­ 3
б­ќSRT П­ 15
W
A
3
B
6
C
x
6.5
F
R
T
7.
6.
12
13
6
X
Z
E
Find x.
5.
S
5
D
7
5
x
9
20
18
x
24
16
8.
PROOF
12
Write a paragraph proof of Theorem 6.7.
Given: б­ќABC Пі б­ќDEF
AB
m
бЋЏбЋЏ П­ бЋЏбЋЏ
DE
n
perimeter of б­ќABC
m
Prove: бЋЏбЋЏбЋЏ П­ бЋЏбЋЏ
n
perimeter of б­ќDEF
Application
B
E
A
C
D
F
9. PHOTOGRAPHY The distance from the film to the lens in a camera is 10
centimeters. The film image is 5 centimeters high. Tamika is 165 centimeters
tall. How far should she be from the camera in order for the photographer to
take a full-length picture?
Lesson 6-5 Parts of Similar Triangles 319
Practice and Apply
For
Exercises
See
Examples
10–15
16, 17, 28
18–27
30–37
1
4
3
2
Find the perimeter of the given triangle.
10. б­ќBCD, if б­ќBCD Пі б­ќFDE, CD П­ 12, 11. б­ќADF, if б­ќADF Пі б­ќBCE, BC П­ 24,
EB П­ 12, CE П­ 18, and DF П­ 21
FD П­ 5, FE П­ 4, and DE П­ 8
F 4
A
E
A
B
C
D
5
8
B
D
Extra Practice
E
12
See page 766.
F
C
12. б­ќCBH, if б­ќCBH Пі б­ќFEH, ADEG is
a parallelogram, CH П­ 7, FH П­ 10,
FE П­ 11, and EH П­ 6
H
7
D
C
E
A
B
C
13. б­ќDEF, if б­ќDEF Пі б­ќCBF, perimeter
of б­ќCBF П­ 27, DF П­ 6, and FC П­ 8
10
F
A
G
D
F
6
B
11
E
14. б­ќABC, if б­ќABC Пі б­ќCBD, CD П­ 4,
DB П­ 3, and CB П­ 5
15. б­ќABC, if б­ќABC Пі б­ќCBD, AD П­ 5,
CD П­ 12, and BC П­ 31.2
C
A
5
4
A
D
5 D
3
B
12
B
31.2
C
16. DESIGN Rosario wants to enlarge the dimensions of an 18-centimeter by
24-centimeter picture by 30%. She plans to line the inside edge of the frame
with blue cord. The store only had 110 centimeters of blue cord in stock. Will
this be enough to fit on the inside edge of the frame? Explain.
17. PHYSICAL FITNESS A park has two
similar triangular jogging paths as shown.
The dimensions of the inner path are
300 meters, 350 meters, and 550 meters.
The shortest side of the outer path is
600 meters. Will a jogger on the inner
path run half as far as one on the
outer path? Explain.
18. Find EG if б­ќACB Пі б­ќEGF, A
а·†D
а·†
is an altitude of б­ќACB, E
а·†H
а·†
is an altitude of б­ќEGF, AC П­ 17,
AD П­ 15, and EH П­ 7.5.
A
17
E
7.5
B
320 Chapter 6 Proportions and Similarity
15
C
D
H G
F
19. Find EH if б­ќABC Пі б­ќDEF, B
а·†G
а·†
is an altitude of б­ќABC, E
а·†H
а·†
is an altitude of б­ќDEF, BG П­ 3,
BF П­ 4, FC П­ 2, and CE П­ 1.
A
G
4
B
20. Find FB if Sа·†A
а·† and Fа·†B
а·† are
altitudes and б­ќRST Пі б­ќEFG.
S
R
A
T
G
K
23.
E
L
M
C
G
14
11
x
20
x–5
2x – 3
25.
4
6
x
6ПЄx
2
32
24.
D
4
x
B
Find x.
22.
12
C1 E
J
7ПЄx
5
E
2
F
21. Find DC if D
а·†G
а·† and Jа·†M
а·† are
altitudes and б­ќKJL Пі б­ќEDC.
F
2
x
D
H
3
x
8
9
2x
xП©3
26. Find UB if б­ќRST Пі б­ќUVW, T
а·†A
а·†
and W
а·†B
а·† are medians, TA П­ 8,
RA П­ 3, WB П­ 3x ПЄ 6, and
UB П­ x П© 2.
27. Find CF and BD if B
а·†Fа·† bisects Р„ABC
and A
а·†C
а·†аїЈ E
а·†D
а·†, BA П­ 6, BC П­ 7.5,
AC П­ 9, and DE П­ 9.
E
A
W
F
T
B
U
R
A
S
B
D
C
V
28. PHOTOGRAPHY One of the first
cameras invented was called a camera
obscura. Light entered an opening in
the front, and an image was reflected
in the back of the camera, upside down,
forming similar triangles. If the image
of the person on the back of the camera
is 12 inches, the distance from the
opening to the person is 7 feet, and
the camera itself is 15 inches long,
how tall is the person being
photographed?
Camera
12 in.
7 ft
29. CRITICAL THINKING C
а·†D
а·† is an altitude to the
hypotenuse A
а·†B
а·†. Make a conjecture about x, y, and z.
Justify your reasoning.
15 in.
A
x
D
z
C
y
B
Lesson 6-5 Parts of Similar Triangles 321
Write the indicated type of proof.
30. a paragraph proof of Theorem 6.8
31. a two-column proof of Theorem 6.10
32. a two-column proof of the Angle
33. a paragraph proof
Bisector Theorem (Theorem 6.11)
Given: б­ќABC Пі б­ќPQR
B
Given: C
а·†D
а·† bisects Р„ACB
а·†D
а·† is an altitude of б­ќABC.
Q
By construction, A
а·†E
а·†аїЈ C
а·†D
а·†.
а·†Sа·† is an altitude of б­ќPQR.
PROOF
QS
QP
Prove: бЋЏбЋЏ П­ бЋЏбЋЏ
AD
BD
Prove: бЋЏбЋЏ П­ бЋЏбЋЏ
AC
BC
BD
BA
E
Q
B
C
3
A
A
1 2
D
C
D
P
B
34. a flow proof
Given: Р„C РҐ Р„BDA
AD
AC
бЋЏ П­ бЋЏбЋЏ
Prove: бЋЏ
BA
DA
C
35. a two-column proof
Given: Jа·†Fа·† bisects Р„EFG.
E
а·†H
а·† аїЈ Fа·†G
а·†, E
а·†Fа·† аїЈ H
а·†G
а·†
EK
GJ
Prove: бЋЏбЋЏ П­ бЋЏбЋЏ
KF
B
JF
J
A
D
K
E
H
F
36. a two-column proof
Given: R
а·†U
а·† bisects Р„SRT;
V
а·†U
а·†аїЈ R
а·†T
а·†.
SV
SR
Prove: бЋЏбЋЏ П­ бЋЏбЋЏ
VR
RT
R
G
37. a flow proof
Given: б­ќRST Пі б­ќABC; W and D are
midpoints of T
а·†Sа·† and C
а·†B
а·†.
Prove: б­ќRWS Пі б­ќADB
A
S
V
R
S
R
U
B
D
C
T
S
W
T
38. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson.
How is geometry related to photography?
Include the following in your answer:
• a sketch of how a camera works showing the image and the film, and
• why the two isosceles triangles are similar.
SOL/EOC Practice
Standardized
Test Practice
39. GRID IN Triangle ABC is similar to б­ќDEF. If AC П­ 10.5, AB П­ 6.5, and DE П­ 8,
find DF.
40. ALGEBRA The sum of three numbers is 180. Two of the numbers are the same,
and each of them is one-third of the greatest number. What is the least number?
A 30
B 36
C 45
D 72
322 Chapter 6 Proportions and Similarity
Maintain Your Skills
Mixed Review
Determine whether M
а·†N
а·†аїЈ O
а·†P
а·†. Justify your
answer. (Lesson 6-4)
41. LM П­ 7, LN П­ 9, LO П­ 14, LP П­ 16
42. LM П­ 6, MN П­ 4, LO П­ 9, OP П­ 6
43. LN П­ 12, NP П­ 4, LM П­ 15, MO П­ 5
O
M
L
P
N
Identify the similar triangles. Find x and the measure(s) of the indicated side(s).
(Lesson 6-3)
44. VW and WX
V
Y
3x – 6
R
6
Q
W
6
45. PQ
5
2x П© 1
xП©4
X
Z
S 4 T
10
P
Write an equation in slope-intercept form for the line that satisfies the given
conditions. (Lesson 3-4)
46. x-intercept is 3, y-intercept is ПЄ3
47. m П­ 2, contains (ПЄ1, ПЄ1)
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Name the next two numbers in each pattern.
(To review patterns, see Lesson 2-1.)
48. 5, 12, 19, 26, 33, …
49. 10, 20, 40, 80, 160, …
P ractice Quiz 2
50. 0, 5, 4, 9, 8, 13, …
Lessons 6-4 and 6-5
Refer to б­ќABC. (Lesson 6-4)
1. If AD П­ 8, AE П­ 12, and EC П­ 18, find AB.
B
2. If AE П­ m ПЄ 2 , EC П­ m П© 4,
AD П­ 4, and AB П­ 20, find m.
VW
Determine whether Y
а·†Z
а·†аїЈ а·†
а·†. Justify your answer. (Lesson 6-4)
3. XY П­ 30, XV П­ 9, XW П­ 12, and XZ П­18
4. XV П­ 34.88, VY П­ 18.32, XZ П­ 33.25, and WZ П­ 11.45
Y
Z
D
A
V
C
E
X
W
Find the perimeter of the given triangle. (Lesson 6-5)
C
5. б­ќDEF if б­ќDEF Пі б­ќGFH
6. б­ќRUW if б­ќRUW Пі б­ќSTV,
G2
H
ST П­ 24, VS П­ 12, VT П­ 18,
2.5
D
4
and UW П­ 21
W
V
F
6
E
Find x. (Lesson 6-5)
7.
14
10
R
8.
x
T
U
9.
4
2x
10
6
18
S
5 x
x
xП©3
10. LANDSCAPING Paulo is designing two gardens shaped like similar triangles.
One garden has a perimeter of 53.5 feet, and the longest side is 25 feet. He wants
the second garden to have a perimeter of 32.1 feet. Find the length of the longest
side of this garden. (Lesson 6-5)
www.geometryonline.com/self_check_quiz/sol
Lesson 6-5 Parts of Similar Triangles 323
A Preview of Lesson 6-6
Sierpinski Triangle
Collect Data
Stage 0 On isometric dot
paper, draw an equilateral
triangle in which each side
is 16 units long.
Stage 0
Stage 0
Stage 1 Connect the
midpoints of each side to
form another triangle. Shade
the center triangle.
Stage 1
Stage 1
Stage 2 Repeat the process
using the three nonshaded
triangles. Connect the
midpoints of each side to form
other triangles.
Stage 2
Stage 2
If you repeat this process indefinitely, the pattern that results is called the
Sierpinski Triangle. Since this figure is created by repeating the same procedure
over and over again, it is an example of a geometric shape called a fractal.
Analyze the Data
1. Continue the process through Stage 4. How many nonshaded triangles do you
have at Stage 4?
2. What is the perimeter of a nonshaded triangle in Stage 0 through Stage 4?
3. If you continue the process indefinitely, describe what will happen to the
perimeter of each nonshaded triangle.
4. Study б­ќDFM in Stage 2 of the Sierpenski Triangle shown at the right.
Is this an equilateral triangle? Are б­ќBCE, б­ќGHL, or б­ќIJN equilateral?
5. Is б­ќBCE Пі б­ќDFM? Explain your answer.
6. How many Stage 1 Sierpinski triangles are there in Stage 2?
A
C
B
E
D
H
G
K
L
M
Make a Conjecture
7. How can three copies of a Stage 2 triangle be combined to form a Stage 3 triangle?
8. Combine three copies of the Stage 4 Sierpinski triangle. Which stage of the
Sierpinski Triangle is this?
9. How many copies of the Stage 4 triangle would you need to make a Stage 6 triangle?
324 Investigating Slope-Intercept Form
324 Chapter 6 Proportions and Similarity
F
I
J
N
P
Fractals and Self-Similarity
• Recognize and describe characteristics of fractals.
• Identify nongeometric iteration.
Vocabulary
• iteration
• fractal
• self-similar
is mathematics found in nature?
Patterns can be found in many objects in nature,
including broccoli. If you take a piece of broccoli
off the stalk, the small piece resembles the whole.
This pattern of repeated shapes at different scales
is part of fractal geometry.
CHARACTERISTICS OF FRACTALS Benoit Mandelbrot, a mathematician,
coined the term fractal to describe things in nature that are irregular in shape, such
as clouds, coastlines, or the growth of a tree. The patterns found in nature are
analyzed and then recreated on a computer, where they can be studied more closely.
These patterns are created using a process called iteration. Iteration is a process of
repeating the same procedure over and over again. A fractal is a geometric figure
that is created using iteration. The pattern’s structure appears to go on infinitely.
By creating a Sierpinski
Triangle, you can find
a pattern in the area
and perimeter of this
well-known fractal.
Visit www.geometry
online.com/webquest
to continue work on
your WebQuest project.
Compare the pictures of a human
circulatory system and the mouth of the
Ganges in Bangladesh. Notice how the
branches of the tributaries have the same
pattern as the branching of the blood
vessels.
One characteristic of fractals is that
they are self similar . That is, the smaller
and smaller details of a shape have the
same geometric characteristics as the
original form.
The Sierpinski Triangle is a fractal that is self-similar. Stage 1 is formed by
drawing the midsegments of an equilateral triangle and shading in the triangle
formed by them. Stage 2 repeats the process in the unshaded triangles. This process
can continue indefinitely with each part still being similar to the original.
Stage 0
Stage 1
Stage 2
Stage 4
The Sierpinski Triangle is said to be strictly self-similar, which means that any of its
parts, no matter where they are located or what size is selected, contain a figure that
is similar to the whole.
Lesson 6-6 Fractals and Self-Similarity 325
(t)C Squared Studios/PhotoDisc, (bl)CNRI/PhotoTake, (br)CORBIS
Example 1 Self-Similarity
Prove that a triangle formed in Stage 2 of a Sierpinski triangle is similar to the
triangle in Stage 0.
The argument will be the same for any triangle in Stage 2, so we will use
only б­ќCGJ from Stage 2.
Given: б­ќABC is equilateral.
D, E, F, G, J, and H are midpoints of
A
а·†B
а·†, B
а·†C
а·†, C
а·†A
а·†, Fа·†C
а·†, C
а·†E
а·†, and Fа·†E
а·†, respectively.
Look Back
To review midsegment,
see Lesson 6-4.
G
F H
A
Prove: б­ќCGJ Пі б­ќCAB
Study Tip
C
J
E
D
B
Statements
1. б­ќABC is equilateral; D, E, F are
midpoints of A
а·†B
а·†, B
а·†C
а·†, C
а·†A
а·†; G,
J, and H are midpoints of Fа·†C
а·†,
C
а·†E
а·†, Fа·†E
а·†.
Reasons
1. Given
2. F
а·†E
а·† is a midsegment of б­ќCAB;
GJа·† is a midsegment of б­ќCFE.
а·†
2. Definition of a Triangle Midsegment
3. Fа·†E
а·†аїЈ A
а·†B
а·†; G
а·†Jа·† аїЈ Fа·†E
а·†
4. G
а·†Jа·† аїЈ A
а·†B
а·†
3. Triangle Midsegment Theorem
5. Р„CGJ РҐ Р„CAB
4. Two segments parallel to the
same segment are parallel.
5. Corresponding Рґ Postulate
6. Р„C РҐ Р„C
6. Reflexive Property
7. б­ќCGJ Пі б­ќCAB
7. AA Similarity
Thus, using the same reasoning, every triangle in Stage 2 is similar to the original
triangle in Stage 0.
You can generate many other fractal images using an iterative process.
Study Tip
Common
Misconceptions
Not all repeated patterns
are self-similar. The Koch
curve is one such example
when applied to a
triangle.
Example 2 Create a Fractal
Draw a segment and trisect it. Create a fractal by replacing the middle third of
the segment with two segments of the same length as the removed segment.
After the first geometric iteration, repeat the
process on each of the four segments in Stage 1.
Stage 5
Continue to repeat the process.
This fractal image is called a Koch curve.
Stage 4
Stage 3
Stage 2
Stage 1
Stage 0
If the first stage is an equilateral triangle, instead of a segment, this iteration will
produce a fractal called Koch’s snowflake.
326
Chapter 6 Proportions and Similarity
NONGEOMETRIC ITERATION An iterative process does not always include
manipulation of geometric shapes. Iterative processes can be translated into
formulas or algebraic equations. These are called recursive formulas.
Study Tip
Recursion on
the Graphing
Calculator
To do recursion on a
graphing calculator, store
2 as the value for X and
press ENTER . Then enter
X 2 : X 2 в†’ X . Press
ENTER for each iteration.
Example 3 Evaluate a Recursive Formula
Find the value of x2, where x initially equals 2. Then use that value as the next x
in the expression. Repeat the process four times and describe your observations.
The iterative process is to square the value repeatedly. Begin with x П­ 2. The value
of x2 becomes the next value for x.
x
2
4
16
256
65,536
x2
4
16
256
65,536
4,294,967,296
The values grow greater with each iteration, approaching infinity.
Example 4 Find a Recursive Formula
PASCAL’S TRIANGLE Pascal’s Triangle is a numerical pattern in which each row
begins and ends with 1 and all other terms in the row are the sum of the two
numbers above it.
a. Find a formula in terms of the row number for the sum of the values in any
row in the Pascal’s triangle.
To find the sum of the values in the tenth row, we can investigate a simpler
problem. What is the sum of values in the first four rows of the triangle?
Row
Pascal’s Triangle
Sum
Pattern
1
1
1
2
1
3
1
4
5
1
1
2
3
4
1
3
6
1
4
1
1
20 П­ 21 ПЄ 1
2
21 П­ 22 ПЄ 1
4
22 П­ 23 ПЄ 1
8
23 П­ 24 ПЄ 1
16
24 П­ 25 ПЄ 1
It appears that the sum of any row is a power of 2. The formula is 2 to a power
that is one less than the row number: An П­ 2n ПЄ 1.
b. What is the sum of the values in the tenth row of Pascal’s triangle?
The sum of the values in the tenth row will be 210 ПЄ 1 or 512.
Example 5 Solve a Problem Using Iteration
BANKING Felisa has $2500 in a money market account that earns 3.2% interest.
If the interest is compounded annually, find the balance of her account after
3 years.
First, write an equation to find the balance after one year.
current balance П© (current balance П« interest rate) П­ new balance
2500 П© (2500 Рё 0.032) П­ 2580
2580 П© (2580 Рё 0.032) П­ 2662.56
2662.56 П© (2662.56 Рё 0.032) П­ 2747.76
After 3 years, Felisa will have $2747.76 in her account.
www.geometryonline.com/extra_examples/sol
Lesson 6-6 Fractals and Self-Similarity 327
Concept Check
1. Describe a fractal in your own words. Include characteristics of fractals in your
answer.
2. Explain why computers provide an efficient way to generate fractals.
3. OPEN ENDED Find an example of fractal geometry in nature, excluding those
mentioned in the lesson.
Guided Practice
For Exercises 4–6, use the following information.
A fractal tree can be drawn by making two new branches from
the endpoint of each original branch, each one-third as long as
the previous branch.
4. Draw Stages 3 and 4 of a fractal tree. How many total
branches do you have in Stages 1 through 4? (Do not
count the stems.)
Stage 1
Stage 2
5. Find a pattern to predict the number of branches at each stage.
6. Is a fractal tree strictly self-similar? Explain.
For Exercises 7–9, use a calculator.
7. Find the square root of 2. Then find the square root of the result.
8. Find the square root of the result in Exercise 7. What would be the result after
100 repeats of taking the square root?
9. Determine whether this is an iterative process. Explain.
Application
10. BANKING Jamir has $4000 in a savings account. The annual percent interest
rate is 1.1%. Find the amount of money Jamir will have after the interest is
compounded four times.
Practice and Apply
For
Exercises
See
Examples
11–13, 21–23
14–20, 25, 28
24–29
30–37
38–40
1, 2
4
2
3
5
Extra Practice
See page 766.
For Exercises 11–13, Stage 1 of a fractal is drawn on grid paper so that each side of
the large square is 27 units long. The trisection points of the sides are connected
to make 9 smaller squares with the middle square shaded. The shaded square is
known as a hole.
11. Copy Stage 1 on your paper. Then draw Stage 2 by
repeating the Stage 1 process in each of the outer eight
squares. How many holes are in this stage?
12. Draw Stage 3 by repeating the Stage 1 process in each
unshaded square of Stage 2. How many holes are in
Stage 3?
13. If you continue the process indefinitely, will the figure you obtain be strictly
self-similar? Explain.
14. Count the number of dots in each
arrangement. These numbers are called
triangular numbers. The second triangular
number is 3 because there are three dots in
the array. How many dots will be in the
seventh triangular number?
328 Chapter 6 Proportions and Similarity
1
3
6
10
For Exercises 15–20, refer to Pascal’s triangle on page 327. Look at the third
diagonal from either side, starting at the top of the triangle.
15. Describe the pattern.
16. Explain how Pascal’s triangle relates to the triangular numbers.
17. Generate eight rows of Pascal’s triangle. Replace each of the even numbers with a
0 and each of the odd numbers with a 1. Color each 1 and leave the 0s uncolored.
Describe the picture.
18. Generate eight rows of Pascal’s triangle. Divide each entry by 3. If the remainder is
1 or 2, shade the number cell black. If the remainder is 0, leave the cell unshaded.
Describe the pattern that emerges.
19. Find the sum of the first 25 numbers in the outside diagonal of Pascal’s triangle.
20. Find the sum of the first 50 numbers in the second diagonal.
The three shaded interior triangles shown were made by
trisecting the three sides of an equilateral triangle and
connecting the points.
21. Prove that one of the nonshaded triangles is similar to the
original triangle.
22. Repeat the iteration once more.
23. Is the new figure strictly self-similar?
Blaise Pascal
Pascal (1623–1662) did not
discover Pascal’s triangle,
but it was named after him
in honor of his work in
1653 called Treatise on the
Arithmetical Triangle. The
patterns in the triangle are
also used in probability.
Source: Great Moments in
Mathematics/After 1650
24. How many nonshaded triangles are in Stages 1 and 2?
Refer to the Koch Curve on page 326.
25. What is a formula for the number of segments in terms of the stage number?
Use your formula to predict the number of segments in Stage 8 of a Koch curve.
26. If the length of the original segment is 1 unit, how long will the segments be in
each of the first four stages? What will happen to the length of each segment as
the number of stages continues to increase?
Refer to the Koch Snowflake on page 326. At Stage 1, the length of each side is
1 unit.
27. What is the perimeter at each of the first four stages of a Koch snowflake?
28. What is a formula for the perimeter in terms of the stage number? Describe the
perimeter as the number of stages continues to increase.
29. Write a paragraph proof to show that the triangles generated on the sides of a
Koch Snowflake in Stage 1, are similar to the original triangle.
Find the value of each expression. Then use that value as the next x in the
expression. Repeat the process four times, and describe your observations.
1
31. бЋЏбЋЏ, where x initially equals 5
30. Н™а·†x, where x initially equals 12
бЋЏ1бЋЏ
32. x 3 , where x initially equals 0.3
x
33. 2x, where x initially equals 0
Find the first three iterates of each expression.
34. 2x П© 1, x initially equals 1
35. x ПЄ 5, where x initially equals 5
36. x2 ПЄ 1, x initially equals 2
37. 3(2 ПЄ x), where x initially equals 4
38. BANKING Raini has a credit card balance of $1250 and a monthly interest rate
of 1.5%. If he makes payments of $100 each month, what will the balance be
after 3 months?
Lesson 6-6 Fractals and Self-Similarity 329
Reunion des Musees Nationaux/Art Resource, NY
WEATHER For Exercises 39 and 40, use the following information.
There are so many factors that affect the weather that it is difficult for meteorologists
to make accurate long term predictions. Edward N. Lorenz called this dependence
the Butterfly Effect and posed the question “Can the flap of a butterfly’s wings in
Brazil cause a tornado in Texas?”
39. Use a calculator to find the first ten iterates of 4x(1 ПЄ x) when x initially equals
0.200 and when the initial value is 0.201. Did the change in initial value affect
the tenth value?
40. Why do you think this is called the Butterfly Effect?
41. ART Describe how artist Jean-Paul Agosti
used iteration and self-similarity in his painting
Jardin du Creuset.
42. NATURE Some of these pictures are of real objects and others are fractal images
of objects.
a. Compare the pictures and identify those you think are of real objects.
b. Describe the characteristics of fractals shown in the images.
flower
mountain
feathers
moss
43. RESEARCH Use the Internet or other sources to find the names and pictures of
the other fractals Waclaw Sierpinski developed.
44. CRITICAL THINKING Draw a right triangle on
grid paper with 6 and 8 units for the lengths of
the perpendicular sides. Shade the triangle
formed by the three midsegments. Repeat the
process for each unshaded triangle. Find the
perimeter of the shaded triangle in Stage 1. What
is the total perimeter of all the shaded triangles
in Stage 2?
Stage 1
6
8
45. WRITING IN MATH
Answer the question that was posed at the beginning
of the lesson.
How is mathematics related to nature?
Include the following in your answer:
• explain why broccoli is an example of fractal geometry, and
• how scientists can use fractal geometry to better understand nature.
330 Chapter 6 Proportions and Similarity
Standardized
Test Practice
SOL/EOC Practice
46. GRID IN A triangle has side lengths of 3 inches, 6 inches, and 8 inches. A
similar triangle is 24 inches on one side. Find the maximum perimeter, in inches,
of the second triangle.
47. ALGEBRA A repair technician charges $80 for the first thirty minutes of each
house call plus $2 for each additional minute. The repair technician charged a
total of $170 for a job. How many minutes did the repair technician work?
A 45 min
B 55 min
C 75 min
D 85 min
Maintain Your Skills
Mixed Review
Find x. (Lesson 6-5)
48.
49.
16
21
3x ПЄ 6
50.
20
14
xП©4
x
51.
3x
6
8
x
17
2x + 1
7
15
x
For Exercises 52–54, refer to ᭝JKL. (Lesson 6-4)
52. If JL П­ 27, BL П­ 9, and JK П­ 18, find JA.
K
A
53. If AB П­ 8, KL П­ 10, and JB П­ 13, find JL.
54. If JA П­ 25, AK П­ 10, and BL П­ 14, find JB.
J
55. FOLKLORE The Bermuda Triangle is
an imaginary region located off the
southeastern Atlantic coast of the
United States. It is the subject of many
stories about unexplained losses of
ships, small boats, and aircraft. Use
the vertex locations to name the
angles in order from least measure to
greatest measure. (Lesson 5-4)
B
L
Bermuda
1042 mi
965 mi
Miami,
Florida
1038 mi
San Juan,
Puerto Rico
Find the length of each side of the polygon for the given perimeter. (Lesson 1-6)
56. P П­ 60 centimeters
57. P П­ 54 feet
58. P П­ 57 units
2n ПЄ 7
nП©2
xП©2
2x П© 1
www.geometryonline.com/self_check_quiz/sol
n
Lesson 6-6 Fractals and Self-Similarity 331
Vocabulary and Concept Check
cross products (p. 283)
extremes (p. 283)
fractal (p. 325)
iteration (p. 325)
means (p. 283)
midsegment (p. 308)
proportion (p. 283)
ratio (p. 282)
scale factor (p. 290)
self-similar (p. 325)
similar polygons (p. 289)
A complete list of postulates and theorems can be found on pages R1–R8.
Exercises State whether each sentence is true or false. If false, replace the
underlined expression to make a true sentence.
1. A midsegment of a triangle is a segment whose endpoints are the
midpoints of two sides of the triangle.
2. Two polygons are similar if and only if their corresponding angles are congruent
and the measures of the corresponding sides are congruent .
3. If two angles of one triangle are congruent to two angles of another triangle, then
the triangles are similar .
4. If two triangles are similar, then the perimeters are proportional to the measures
of the corresponding angles .
5. A fractal is a geometric figure that is created using recursive formulas .
6. A midsegment of a triangle is parallel to one side of the triangle, and its length is
twice the length of that side.
a
c
7. For any numbers a and c and any nonzero numbers b and d, бЋЏбЋЏ П­ бЋЏбЋЏ if and only if
b
d
ad П­ bc .
8. If two triangles are similar, then the measures of the corresponding angle bisectors
of the triangle are proportional to the measures of the corresponding sides .
9. If a line intersects two sides of a triangle and separates the sides into
corresponding segments of proportional lengths, then the line is
equal to one-half the length of the third side.
6-1 Proportions
See pages
282–287.
Example
Concept Summary
• A ratio is a comparison of two quantities.
• A proportion is an equation stating that two ratios are equal.
z
40
z
5
бЋЏбЋЏ П­ бЋЏбЋЏ
40
8
5
8
Solve бЋЏбЋЏ П­ бЋЏбЋЏ.
Original proportion
z Рё 8 П­ 40(5) Cross products
8z П­ 200
z П­ 25
332 Chapter 6 Proportions and Similarity
Multiply.
Divide each side by 8.
www.geometryonline.com/vocabulary_review
Chapter 6 Study Guide and Review
Exercises
Solve each proportion. See Example 3 on page 284.
xП©2
14
12. бЋЏбЋЏ П­ бЋЏбЋЏ
x
3
10. бЋЏбЋЏ П­ бЋЏбЋЏ
12
4
7
28
11. бЋЏбЋЏ П­ бЋЏбЋЏ
7
3
13. бЋЏбЋЏ П­ бЋЏбЋЏ
4ПЄx
16
14. бЋЏбЋЏ П­ бЋЏбЋЏ
3
yПЄ3
7
z
3П©x
5
10
x ПЄ 12
xП©7
15. бЋЏбЋЏ П­ бЋЏбЋЏ
25
ПЄ4
6
16. BASEBALL A player’s slugging percentage is the ratio of the number of total
bases from hits to the number of total at-bats. The ratio is converted to a decimal
(rounded to three places) by dividing. If Alex Rodriguez of the Texas Rangers has
263 total bases in 416 at-bats, what is his slugging percentage?
17. A 108-inch-long board is cut into two pieces that have lengths in the ratio 2:7.
How long is each new piece?
6-2 Similar Polygons
See pages
289–297.
Example
Concept Summary
• In similar polygons, corresponding angles are congruent, and
corresponding sides are in proportion.
• The ratio of two corresponding sides in two similar polygons is the scale
factor.
Determine whether the pair of triangles
is similar. Justify your answer.
Р„AРҐР„D and Р„C РҐ Р„F, so by the Third
Angle Theorem, Р„B РҐ Р„E. All of the
corresponding angles are congruent.
A
10
16
5
4
8.8 E
16
CA
бЋЏбЋЏ П­ бЋЏбЋЏ
12.8
FD
5
4
П­ бЋЏбЋЏ or 1.25
F
C
11
BC
бЋЏбЋЏ П­ бЋЏбЋЏ
8. 8
EF
8
12.8
B
11
Now, check to see if corresponding sides
are in proportion.
AB
10
бЋЏбЋЏ П­ бЋЏбЋЏ
DE
8
D
5
4
П­ бЋЏбЋЏ or 1.25
П­ бЋЏбЋЏ or 1.25
The corresponding angles are congruent, and the ratios of the measures of the
corresponding sides are equal, so б­ќABC Пі б­ќDEF.
Exercises Determine whether each pair of figures is similar. Justify your answer.
See Example 1 on page 290.
18. T
6
19.
U
L
30
M
24
N
K
V
9
P
20
16
W
R
Q
Chapter 6 Study Guide and Review 333
Chapter 6 Study Guide and Review
Each pair of polygons is similar. Write a similarity statement, and find x, the
measures of the indicated sides, and the scale factor. See Example 3 on page 291.
20. A
B and A
G
21. P
Q and Q
S
а·†а·†
а·†а·†
а·†а·†
а·†а·†
S
E
F
3П©x
5
B
P
C
6ПЄx
Q
xПЄ2
A
x
3
7.5
D
G
6П©x
R
T
6-3 Similar Triangles
See pages
298–306.
Example
Concept Summary
• AA, SSS, and SAS Similarity can all be used to prove triangles similar.
• Similarity of triangles is reflexive, symmetric, and transitive.
INDIRECT MEASUREMENT Alonso wanted to determine the height of a tree on
the corner of his block. He knew that a certain fence by the tree was 4 feet tall.
At 3 P.M., he measured the shadow of the fence to be 2.5 feet tall. Then he measured
the tree’s shadow to be 11.3 feet. What is the height of the tree?
Since the triangles formed are similar, a proportion
can be written. Let x be the height of the tree.
height of the tree
height of the fence
П­
tree shadow length
fence shadow length
x
11.3
бЋЏбЋЏ П­ бЋЏбЋЏ
4
2.5
x Рё2.5 П­ 4(11.3)
2.5x П­ 45.2
x П­ 18.08
Substitution
Cross products
4 ft
Simplify.
11.3 ft
2.5 ft
Divide each side by 2.5.
The height of the tree is 18.08 feet.
Exercises Determine whether each pair of triangles is similar. Justify your answer.
See Example 1 on page 299.
22.
23.
A
24. L
G
D
B
C
F
J
E
Identify the similar triangles. Find x.
25. B
xП©3
N
35Лљ
H
D
I
M
85Лљ
K
40Лљ
Q
P
See Example 2 on page 300.
26. R
2x
6
S
1
A
C
4
11x ПЄ 2
E
334 Chapter 6 Proportions and Similarity
V
3
U
xП©2
T
Chapter 6 Study Guide and Review
6-4 Parallel Lines and Proportional Parts
See pages
307–315.
Example
Concept Summary
• A segment that intersects two sides of a triangle and is parallel to the third
side divides the two intersected sides in proportion.
• If two lines divide two segments in proportion, then the lines are parallel.
In б­ќTRS, TS П­ 12. Determine whether а·†
MN
а·† аїЈ Sа·†R
а·†.
If TS П­ 12, then MS П­ 12 ПЄ 9 or 3. Compare the
segment lengths to determine if the lines are parallel.
TM
9
бЋЏбЋЏ П­ бЋЏбЋЏ П­ 3
MS
3
TM
Because бЋЏбЋЏ
MS
Exercises
T
9
10
M
TN
10
бЋЏбЋЏ П­ бЋЏбЋЏ П­ 2
NR
5
TN
бЋЏбЋЏ, M
а·†N
а·† НћаїЈ Sа·†R
а·†.
NR
N
5
S
R
Determine whether G
а·†L
а·†аїЈ H
а·†K
а·†. Justify your answer.
G
See Example 2 on page 308.
H
27. IH П­ 21, HG П­ 14, LK П­ 9, KI П­ 15
28. GH П­ 10, HI П­ 35, IK П­ 28, IL П­ 36
L
29. GH П­ 11, HI П­ 22, and IL is three times the length of K
а·†Lа·†.
K
I
30. LK П­ 6, KI П­ 18, and IG is three times the length of H
а·†Iа·†.
C
Refer to the figure at the right. See Example 1 on page 308.
31. Find ED if AB П­ 6, BC П­ 4, and AE П­ 9.
B
D
E
32. Find AE if AB П­ 12, AC П­ 16, and ED П­ 5.
33. Find CD if AE П­ 8, ED П­ 4, and BE П­ 6.
A
34. Find BC if BE П­ 24, CD П­ 32, and AB П­ 33.
6-5 Parts of Similar Triangles
See pages
316–323.
Example
Concept Summary
• Similar triangles have perimeters proportional to the corresponding sides.
• Corresponding angle bisectors, medians, and altitudes of similar triangles
have lengths in the same ratio as corresponding sides.
If Fа·†B
а·†аїЈ E
а·†C
а·†, A
а·†D
а·† is an angle bisector of Р„A, BF П­ 6, CE П­ 10, and AD П­ 5, find AM.
By AA Similarity using Р„AFE РҐ Р„ABF and Р„A РҐ Р„A, б­ќABF Пі б­ќACE.
BF
AM
бЋЏбЋЏ П­ бЋЏбЋЏ
CE
AD
x
6
бЋЏбЋЏ П­ бЋЏбЋЏ
5
10
10x П­ 30
xП­3
Піб­ќs have angle bisectors in the same
proportion as the corresponding sides.
A
F
AD П­ 5, AF П­ 6, FE П­ 4, AM П­ x
Cross products
E
M
D
B
C
Divide each side by 10.
Thus, AM П­ 3.
Chapter 6 Study Guide and Review 335
• Extra Practice, see pages 764–766.
• Mixed Problem Solving, see page 787.
Exercises Find the perimeter of the given triangle. See Example 1 on page 316.
35. б­ќDEF if б­ќDEF Пі б­ќABC
36. б­ќQRS if б­ќQRS Пі б­ќQTP
P
D
11
A
7
3
C
Q5
R
16
6
B
F
9
15
E
S
T
37. б­ќCPD if the perimeter of б­ќBPA is 12,
BM П­ Н™13
а·†, and CN П­ 3Н™13
а·†
38. б­ќPQR, if б­ќPQM Пі б­ќPRQ
Q
C
13
A
M
P
N
P
D
12
M
R
B
6-6 Fractals and Self-Similarity
See pages
325–331.
Example
Concept Summary
• Iteration is the creation of a sequence by repetition of the same operation.
• A fractal is a geometric figure created by iteration.
• An iterative process involving algebraic equations is a recursive formula.
x
Find the value of бЋЏбЋЏ П© 4, where x initially equals ПЄ8. Use that value as the next
2
x in the expression. Repeat the process five times and describe your observations.
Make a table to organize each iteration.
Iteration
1
2 3 4
5
6
x
ПЄ8
0
4
6
7
7.5
x
бЋЏбЋЏ П©4
2
0
4
6
7
7.5
7.75
The x values appear to get closer to the number 8 with each iteration.
Exercises Draw Stage 2 of the fractal shown below. Determine whether Stage 2
is similar to Stage 1. See Example 2 on page 326.
39.
Stage 0
Stage 1
Find the first three iterates of each expression. See Example 3 on page 327.
40. x3 ПЄ 4, x initially equals 2
41. 3x П© 4, x initially equals ПЄ4
1
42. бЋЏбЋЏ, x initially equals 10
x
336 Chapter 6 Proportions and Similarity
x
43. бЋЏбЋЏ ПЄ 9, x initially equals 30
10
Vocabulary and Concepts
Choose the answer that best matches each phrase.
1. an equation stating that two ratios are equal
2. the ratio between corresponding sides of two similar figures
3. the means multiplied together and the extremes multiplied together
a. scale factor
b. proportion
c. cross products
Skills and Applications
Solve each proportion.
x
1
4. бЋЏбЋЏ П­ бЋЏбЋЏ
14
kП©2
kПЄ2
6. бЋЏбЋЏ П­ бЋЏбЋЏ
4x
108
5. бЋЏбЋЏ П­ бЋЏбЋЏ
2
3
x
7
3
Each pair of polygons is similar. Write a similarity statement and find the scale factor.
A
B
2x П© 2
7.
8. P
9. A
Q 6
S
55Лљ
C
12
F
2x
D
E
145Лљ
15
x П© 20
T
ПЄ3x
G
55Лљ
15
12
M
R
18
B
25
C
151Лљ
H
I
D
Determine whether each pair of triangles is similar. Justify your answer.
10.
11.
12.
M
Q
A
P
6
12
10
5
P
R 3 Q
N
6
E
T
62Лљ
L
S
66Лљ
C
G
Find the perimeter of the given triangle.
16. б­ќDEF, if б­ќDEF Пі б­ќACB
D
K
H
J
I
17. б­ќABC
A
10
A
13
C 10 B
6
F
B
R
Refer to the figure at the right.
13. Find KJ if GJ П­ 8, GH П­ 12, and HI П­ 4.
14. Find GK if GI П­ 14, GH П­ 7, and KJ П­ 6.
15. Find GI if GH П­ 9, GK П­ 6, and KJ П­ 4.
7
D
14
M
E
C
B
18. Find the first three iterates of 5x П© 27 when x initially equals ПЄ3.
19. BASKETBALL Terry wants to measure the height of the top of the backboard of his
basketball hoop. At 4:00, the shadow of a 4-foot fence is 20 inches, and the shadow
of the backboard is 65 inches. What is the height of the top of the backboard?
20. STANDARDIZED TEST PRACTICE If a person’s weekly salary is $X and $Y
is saved, what part of the weekly salary is spent?
SOL/EOC
Practice
A
X
бЋЏбЋЏ
Y
B
XПЄY
бЋЏбЋЏ
X
www.geometryonline.com/chapter_test /sol
C
XПЄY
бЋЏбЋЏ
Y
D
YПЄX
бЋЏбЋЏ
Y
Chapter 6 Practice Test 337
SOL/EOC Practice
5. Miguel is using centimeter grid paper to make
a scale drawing of his favorite car. Miguel’s
drawing is 11.25 centimeters wide. How many
feet long is the actual car? (Lesson 6-1)
Part 1 Multiple Choice
Record your answers on the answer sheet
provided by your teacher or on a sheet of
paper.
1. Which of the following is equivalent to
пЈ¦ПЄ8 П© 2пЈ¦? (Prerequisite Skill)
A
10
B
6
C
ПЄ6
D
ПЄ10
2. Kip’s family moved to a new house. He used a
coordinate plane with units in miles to locate
his new house and school in relation to his old
house. What is the distance between his new
house and school? (Lesson 1-3)
A
12 miles
B
а·† miles
Н™229
C
17 miles
D
а·† miles
Н™425
y
new house
(ПЄ3, 17)
school
(ПЄ5, 2)
old house
O
A
15.0 ft
B
18.75 ft
C
22.5 ft
D
33.0 ft
scale: 1.5 cm = 2 ft
6. Joely builds a large corkboard for her room
that is 45 inches tall and 63 inches wide. She
wants to build a smaller corkboard with a
similar shape for the kitchen. Which of the
following could be the dimensions of that
corkboard? (Lesson 6-2)
A
4 in. by 3 in.
B
7 in. by 5 in.
C
12 in. by 5 in.
D
21 in. by 14 in.
x
7. If б­ќPQR and б­ќSTU are similar, which of the
following is a correct proportion? (Lesson 6-3)
3. The diagonals of rectangle ABCD are A
а·†C
а·† and
BD
а·†
а·†. Hallie found that the distances from the
point where the diagonals intersect to each
vertex were the same. Which of the following
conjectures could Hallie make? (Lesson 2-1)
A
Diagonals of a rectangle are congruent.
B
Diagonals of a rectangle create equilateral
triangles.
C
Diagonals of a rectangle intersect at more
than one point.
D
Diagonals of a rectangle are congruent to
the width.
4. If two sides of a triangular sail are congruent,
which of the following terms cannot be used to
describe the shape of the sail? (Lesson 4-1)
A
acute
B
equilateral
C
obtuse
D
scalene
338 Chapter 6 Proportion and Similarity
A
B
C
D
s
бЋЏбЋЏ
u
s
бЋЏбЋЏ
u
s
бЋЏбЋЏ
u
s
бЋЏбЋЏ
u
t
q
p
П­ бЋЏбЋЏ
q
p
П­ бЋЏбЋЏ
r
r
П­ бЋЏбЋЏ
p
П­ бЋЏбЋЏ
R
U
q
p
r
P
t
Q
s
u
S
T
8. In б­ќABC, D is the midpoint of A
а·†B
а·†, and E is the
midpoint of A
а·†C
а·†. Which of the following is not
true? (Lesson 6-4)
A
AD
AE
бЋЏбЋЏ П­ бЋЏбЋЏ
DB
EC
B
BC
D
EаїЈа·†
а·†а·†
а·†
A
2
C
б­ќABC Пі б­ќADE
D
Р„1 РҐ Р„4
1
D
3
B
E
4
C
Preparing for Standardized Tests
For test-taking strategies and more
practice, see pages 795– 810.
Part 2 Short Response/Grid In
Record your answers on the answer sheet
provided by your teacher or on a sheet of
paper.
9. During his presentation, Dante showed a
picture of several types of balls used in
sports. From this picture, he conjectured that
all balls used in sports are spheres. Brianna
then showed another ball. What is this type
of example called? (Lesson 2-1)
Dante
Brianna
Test-Taking Tip
Question 7
In similar triangles, corresponding angles are congruent
and corresponding sides are proportional. When you set
up a proportion, be sure that it compares corresponding
sides. In this question, p corresponds to s, q corresponds
to t, and r corresponds to u.
Part 3 Extended Response
Record your answers on a sheet of paper.
Show your work.
13. A cable company charges a one-time
connection fee plus a monthly flat rate as
shown in the graph.
Reliable Cable Company
10. What is an equation of the line with slope 3
that contains A(2, 2)? (Lesson 3-4)
Total Amount Paid
(dollars)
11. In б­ќDEF, P is the midpoint
of D
а·†E
а·†, and Q is the midpoint
of side D
а·†Fа·†. If EF П­ 3x П© 4
and PQ П­ 20, what is the
value of x? (Lesson 6-4)
210
F
Q
180
150
120
90
60
30
0
D
P
1
2
3
4
a. What is the slope of the line that joins the
points on the graph? (Lesson 3-3)
E
b. Discuss what the value of the slope
represents. (Lesson 3-3)
12. A city planner designs a triangular traffic
median on Main Street to provide more
green space in the downtown area. The
planner builds a model so that the section
of the median facing Main Street East
measures 20 centimeters. What is the
perimeter, in centimeters, of the model of
the traffic median? (Lesson 6-5)
c. Write an equation of the line. (Lesson 3-4)
d. If the company presents a special offer
that lowers the monthly rate by $5, how
will the equation and graph change?
(Lesson 3-4)
14. Given б­ќADE and а·†
Bа·†
C is equidistant from а·†
Dа·†
E.
Ma
in S
tre
et W
a. Prove that б­ќABC Пі б­ќADE.
est
23 m
46 m
40 m
Main Street East
www.geometryonline.com/standardized_test/sol
(Lessons 6-3 and 6-4)
b. Suppose AB П­ 3500 feet, BD П­ 1500 feet,
and BC П­ 1400 feet. Find DE. (Lesson 6-3)
Chapter 6 Standardized Test Practice 339
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